Lógica/Cálculo Proposicional Clássico/Dedução Natural

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  <page>
    <title>Lógica/Cálculo Proposicional Clássico/Dedução Natural - Parte II/Resolução dos Exercícios</title>
    <ns>0</ns>
    <id>10067</id>
    <revision>
      <id>500219</id>
      <parentid>473170</parentid>
      <timestamp>2025-06-22T18:31:24Z</timestamp>
      <contributor>
        <ip>2804:5C14:8018:5E00:8D6C:C04F:D292:8333</ip>
      </contributor>
      <comment>/* 11 */</comment>
      <origin>500219</origin>
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      <text bytes="50278" sha1="1gpjkdnuo8r4dhi7b11t3ydk5s8imr1" xml:space="preserve">==Resolução de algumas [[Lógica/Cálculo Proposicional Clássico/Dedução Natural - Parte II#Regras de Inferência Derivadas|Regras de Inferência Derivadas]]==
====[[Lógica/Cálculo Proposicional Clássico/Dedução Natural - Parte II#Contraposição (CT)|Contraposição]]====
&lt;math&gt;\neg \beta \to \neg \alpha \vdash \alpha \to \beta&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 1.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\neg \beta \to \neg \alpha&lt;/math&gt;
| &amp;nbsp;
| '''Premissa'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 2. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\alpha&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
| '''Hipótese'''
|-
| 3.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \neg \alpha&lt;/math&gt;
| &amp;nbsp;
| '''2 DN''' 
|-
| 4.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \neg \beta&lt;/math&gt;
| &amp;nbsp;
| '''1,3 MT'''
|-
| 5.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\beta&lt;/math&gt;
| &amp;nbsp;
| '''4 DN'''
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 6.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\alpha \to \beta&lt;/math&gt;
| &amp;nbsp;
| '''2,5 RPC'''
|}
|}

====[[Lógica/Cálculo Proposicional Clássico/Dedução Natural - Parte II#Lei de Duns Scot (DS)|Lei de Duns Scot]]====
&lt;math&gt;\alpha\vdash \neg \alpha\to\beta &lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 1.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\alpha&lt;/math&gt;
| &amp;nbsp;
| '''Premissa'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 2. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\neg \alpha&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
| '''Hipótese''' 
|-
| 3.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\beta&lt;/math&gt;
| &amp;nbsp;
| '''1,2 CTR''' 
|-
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 4.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \alpha\to\beta&lt;/math&gt;
| &amp;nbsp;
| '''2,3 RPC'''
|}
|}

====[[Lógica/Cálculo Proposicional Clássico/Dedução Natural - Parte II#Lei De Morgan I (DM)|Lei De Morgan I]]====
&lt;math&gt;\neg \alpha \land \neg \beta \vdash \neg \left( \alpha \lor \beta \right)&lt;/math&gt;

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| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 1.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\neg \alpha \land \neg \beta&lt;/math&gt;
| &amp;nbsp;
| '''Premissa'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 2. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\alpha \lor \beta&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
| '''Hipótese'''
|-
| 3.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \alpha&lt;/math&gt;
| &amp;nbsp;
| '''1 S''' 
|-
| 4.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \beta&lt;/math&gt;
| &amp;nbsp;
| '''1 S''' 
|-
| 5.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\beta&lt;/math&gt;
| &amp;nbsp;
| '''2,3 SD''' 
|-
| 6.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\beta \land \neg \beta&lt;/math&gt;
| &amp;nbsp;
| '''5,4 C'''
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 7.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \left( \alpha \lor \beta \right)&lt;/math&gt;
| &amp;nbsp;
| '''2,6 RAA'''
|}
|}

====[[Lógica/Cálculo Proposicional Clássico/Dedução Natural - Parte II#Lei De Morgan II (DM)|Lei De Morgan II]]====
&lt;math&gt;\neg \alpha \lor \neg \beta\vdash \neg \left(\alpha\land \beta\right)&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 1.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\neg \alpha \lor \neg \beta&lt;/math&gt;
| &amp;nbsp;
| '''Premissa'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 2. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\alpha \land \beta&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
| '''Hipótese'''
|-
| 3.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\alpha&lt;/math&gt;
| &amp;nbsp;
| '''2 S''' 
|-
| 4.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg\neg \alpha&lt;/math&gt;
| &amp;nbsp;
| '''3 DN''' 
|-
| 5.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg\beta&lt;/math&gt;
| &amp;nbsp;
| '''1,4 SD''' 
|-
| 6.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\beta&lt;/math&gt;
| &amp;nbsp;
| '''2 S'''
|-
| 7.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\beta\land \neg \beta&lt;/math&gt;
| &amp;nbsp;
| '''5,6 C'''
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 8.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \left(\alpha\land \beta\right)&lt;/math&gt;
| &amp;nbsp;
| '''2,7 RAA'''
|}
|}

==Resolução dos [[Lógica/Cálculo Proposicional Clássico/Dedução Natural - Parte II#Exercícios|Exercícios de Aplicação das Regras de Inferência Derivadas]]==
====1====
&lt;math&gt;\left \{\left(A\land B\right)\to C , \neg C\right \}\vdash \neg A \lor \neg B&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 1.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\left(A\land B\right)\to C&lt;/math&gt;
| &amp;nbsp;
| '''Premissa'''
|-
| 2.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\neg C&lt;/math&gt;
| &amp;nbsp;
| '''Premissa'''
|- style=&quot;height:10px&quot;
| 3.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \left(A\land B\right)&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
| '''1,2 MT'''
|-
| 4.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg A \lor \neg B&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
| '''3 DM'''
|}
|}

====2====
&lt;math&gt;\neg \left(A\land \neg B\right)\vdash A\to B&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 1.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\neg \left(A\land \neg B\right)&lt;/math&gt;
| &amp;nbsp;
| '''Premissa'''
|-
| 2.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg A\lor \neg\neg B&lt;/math&gt;
| &amp;nbsp;
| '''1 DM'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 3. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;A&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
| '''Hipótese''' 
|-
| 4.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \neg A&lt;/math&gt;
| &amp;nbsp;
| '''3 DN''' 
|-
| 5.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \neg B&lt;/math&gt;
| &amp;nbsp;
| '''2,4 SD''' 
|-
| 6.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt; B&lt;/math&gt;
| &amp;nbsp;
| '''5 DN''' 
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 7.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A\to B&lt;/math&gt;
| &amp;nbsp;
| '''3,6 RPC'''
|}
|}

====3====
&lt;math&gt;\neg A\to B\vdash A\lor B&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 1.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\neg A\to B&lt;/math&gt;
| &amp;nbsp;
|  &amp;nbsp;  &amp;nbsp;  &amp;nbsp;'''Premissa'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 2. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\neg \left(A\lor B \right)&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
| '''Hipótese''' 
|-
| 3.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg A \land \neg B&lt;/math&gt;
| &amp;nbsp;
| '''2 DM''' 
|-
| 4.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg A&lt;/math&gt;
| &amp;nbsp;
| '''3 S''' 
|-
| 5.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg B&lt;/math&gt;
| &amp;nbsp;
| '''3 S''' 
|-
| 6.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt; B&lt;/math&gt;
| &amp;nbsp;
| '''1,4 MP''' 
|-
| 7.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt; B \land \neg B&lt;/math&gt;
| &amp;nbsp;
| '''6,5 C''' 
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 8.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \neg \left(A\lor B \right) &lt;/math&gt;
| &amp;nbsp;
|  &amp;nbsp;  &amp;nbsp; '''2,7 RAA'''
|- 
| 9.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A\lor B&lt;/math&gt;
| &amp;nbsp;
|  &amp;nbsp;  &amp;nbsp; '''8 DN'''
|}
|}

====4====
&lt;math&gt;\left\{A\to C , B\to C\right\}\vdash \left(A\lor B\right)\to C&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 1.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;A\to C&lt;/math&gt;
| &amp;nbsp;
|  &amp;nbsp;  &amp;nbsp;  &amp;nbsp;'''Premissa'''
|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 2.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;B\to C&lt;/math&gt;
| &amp;nbsp;
|  &amp;nbsp;  &amp;nbsp;  &amp;nbsp;'''Premissa'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 3. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\neg C&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
| '''Hipótese''' 
|-
| 4.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg A&lt;/math&gt;
| &amp;nbsp;
| '''1,3 MT''' 
|-
| 5.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg B&lt;/math&gt;
| &amp;nbsp;
| '''2,3 MT''' 
|-
| 6.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg A \land \neg B&lt;/math&gt;
| &amp;nbsp;
| '''4,5 C''' 
|-
| 7.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt; \neg \left(A \lor B\right)&lt;/math&gt;
| &amp;nbsp;
| '''6 DM''' 
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 8.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg C \to \neg \left(A\lor B \right) &lt;/math&gt;
| &amp;nbsp;
|  '''3,7 RPC'''
|- 
| 9.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\left(A\lor B \right)\to C&lt;/math&gt;
| &amp;nbsp;
|  '''8 CT'''
|}
|}

====5====
&lt;math&gt;\neg \left(A\to B\right)\vdash A\land \neg B&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 01.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\neg \left(A\to B\right)&lt;/math&gt;
| &amp;nbsp;
|  &amp;nbsp; &amp;nbsp; &amp;nbsp;'''Premissa'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 02. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\neg \left( A \land \neg B\right) &lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
| &amp;nbsp; &amp;nbsp; '''Hipótese'''
|- 
| 03. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg A \lor \neg \neg B &lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
| &amp;nbsp; &amp;nbsp; '''2 DM'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 04.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;A&lt;/math&gt;
| &amp;nbsp;
| '''Hipótese''' 
| &amp;nbsp;
|-
| 05.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \neg A&lt;/math&gt;
| &amp;nbsp;
| '''4 DN''' 
|-
| 06.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \neg B&lt;/math&gt;
| &amp;nbsp;
| '''3,5 SD''' 
|-
| 07.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt; B&lt;/math&gt;
| &amp;nbsp;
| '''6 DN'''
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 08.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A\to B&lt;/math&gt;
| &amp;nbsp;
| &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; '''4,7 RPC''' 
|-
| 09.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\left(A\to B\right)\land \neg \left(A\to B\right)&lt;/math&gt;
| &amp;nbsp;
| &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; '''8,1 C''' 
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 10. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \neg \left( A \land \neg B \right)&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
|  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; '''2,9 RAA'''
|-
| 11. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A \land \neg B&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
|  &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; &amp;nbsp; '''10 DN'''
|}
|}

====6====
&lt;math&gt;A\to B\vdash \left(A\lor C\right)\to \left(B\lor C\right)&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 01.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;A\to B&lt;/math&gt;
| &amp;nbsp;
|  &amp;nbsp; &amp;nbsp; &amp;nbsp;'''Premissa'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 02. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;A\lor C&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
| &amp;nbsp; &amp;nbsp; '''Hipótese'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 03.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\neg\left(B\lor C\right)&lt;/math&gt;
| &amp;nbsp;
| '''Hipótese''' 
| &amp;nbsp;
|-
| 04.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg B\land \neg C&lt;/math&gt;
| &amp;nbsp;
| '''3 DM''' 
|-
| 05.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg C&lt;/math&gt;
| &amp;nbsp;
| '''4 S''' 
|-
| 06.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg B&lt;/math&gt;
| &amp;nbsp;
| '''4 S'''
|-
| 07.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A&lt;/math&gt;
| &amp;nbsp;
| '''2,5 SD'''
|-
| 08.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;B&lt;/math&gt;
| &amp;nbsp;
| '''1,7 MP'''
|-
| 09.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;B\land \neg B&lt;/math&gt;
| &amp;nbsp;
| '''1,7 MP'''
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 10.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \neg \left(B\lor C\right)&lt;/math&gt;
| &amp;nbsp;
| &amp;nbsp; &amp;nbsp; &amp;nbsp;'''3,9 RAA''' 
|-
| 11.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;B\lor C&lt;/math&gt;
| &amp;nbsp;
| &amp;nbsp; &amp;nbsp; &amp;nbsp;'''10 DN''' 
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| 12. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\left(A\lor C\right)\to \left(B\lor C\right)&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
|  '''2,11 RPC'''
|}
|}

==Resolução dos Exercícios de Demonstração de [[Lógica/Cálculo Proposicional Clássico/Dedução Natural - Parte II#Teoremas|Teoremas]]==
====1====
&lt;math&gt;\vdash A\leftrightarrow A&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 1. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;  &amp;nbsp; &amp;nbsp;  &amp;nbsp; '''Hipótese'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 2.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|&lt;math&gt;A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;  &amp;nbsp; &amp;nbsp;  &amp;nbsp; '''1 R'''
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 3.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A\to A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''1,2 RPC'''
|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 4.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A\leftrightarrow A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''3,3 CB'''
|}
|}

====2====
&lt;math&gt;\vdash A\leftrightarrow \neg \neg A&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 1. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
|   &amp;nbsp; &amp;nbsp; &amp;nbsp; '''Hipótese'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 2.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|&lt;math&gt;\neg \neg A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
|   &amp;nbsp; &amp;nbsp; &amp;nbsp; '''1 DN'''
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 3.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A\to \neg \neg A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''1,2 RPC'''
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 4. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\neg \neg A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
|  &amp;nbsp; &amp;nbsp;  &amp;nbsp; '''Hipótese'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 5.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|&lt;math&gt;A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
|  &amp;nbsp; &amp;nbsp;  &amp;nbsp; '''4 DN'''
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 6.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \neg A\to A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''4,5 RPC'''
|-
| width=&quot;15px&quot; | 7.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A\leftrightarrow \neg \neg A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''3,6 CB'''
|}
|}

====3====
&lt;math&gt;\vdash \neg\left(A\leftrightarrow \neg A\right)&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 01. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;A\leftrightarrow \neg A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;  &amp;nbsp; &amp;nbsp;  &amp;nbsp; '''Hipótese'''
|
|- 
| 02.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A\to \neg A&lt;/math&gt;
| &amp;nbsp;
| &amp;nbsp;  &amp;nbsp; &amp;nbsp;  '''1 BC'''
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 03. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;A&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
|
| '''Hipótese'''
|-
| 04.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg A&lt;/math&gt;
| &amp;nbsp;
|
| '''2,3 MP'''
|- 
| 05.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A\land \neg A&lt;/math&gt;
| &amp;nbsp;
| &amp;nbsp;
| '''3,4 C'''
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 06.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;
|
| '''3,5 RAA'''
|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 07.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg A\to A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;
|
| '''1 BC'''
|- 
| 08.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A&lt;/math&gt;
| &amp;nbsp;
| &amp;nbsp;
|
| '''7,6 MP'''
|- 
| 09.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A \land \neg A&lt;/math&gt;
| &amp;nbsp;
| &amp;nbsp;
|
| '''8,6 C'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 10.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg\left(A\leftrightarrow \neg A\right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''1,9 RAA'''
|}
|}

====4====
&lt;math&gt;\vdash A\to\left(B\to A\right)&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 1. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;  &amp;nbsp; &amp;nbsp;  &amp;nbsp; '''Hipótese'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 2. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;B&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
|
| '''Hipótese'''
|-
| 3.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt; A&lt;/math&gt;
| &amp;nbsp;
|
| '''1 R'''
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 4.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;B \to A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;
|
| '''2,3 RPC'''
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 5.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A\to\left(B\to A\right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''1,4 RPC'''
|}
|}

====5====
&lt;math&gt;\vdash \left(\neg A\to A\right)\to A&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 1. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\neg A\to A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;  &amp;nbsp; &amp;nbsp;  &amp;nbsp; '''Hipótese'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 2. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\neg A&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
|
| '''Hipótese'''
|-
| 3.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt; A&lt;/math&gt;
| &amp;nbsp;
|
| '''1,2 MP'''
|-
| 4.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt; A\land \neg A&lt;/math&gt;
| &amp;nbsp;
|
| '''3,2 C'''
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 5.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \neg A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;
|
| &amp;nbsp; &amp;nbsp; &amp;nbsp; '''2,4 RAA'''
|- 
| width=&quot;15px&quot; | 6.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;
|
| &amp;nbsp; &amp;nbsp; &amp;nbsp; '''5 DN'''
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 7.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\left(\neg A\to A\right)\to A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''1,6 RPC'''
|}
|}

====6====
&lt;math&gt;\vdash P\to \left(Q\to \left(P\land Q\right)\right)&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 1. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;P&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;  &amp;nbsp; &amp;nbsp;  &amp;nbsp; '''Hipótese'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 2. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;Q&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
|
| '''Hipótese'''
|-
| 3.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt; P\land Q&lt;/math&gt;
| &amp;nbsp;
|
| '''1,2 C'''
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 4.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;Q\to \left(P\land Q\right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;
|
| '''2,3 RPC'''
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 5.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;P\to \left(Q\to \left(P\land Q\right)\right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''1,4 RPC'''
|}
|}

====7====
&lt;math&gt;\vdash \left(\left(A\land B\right)\to C\right)\leftrightarrow \left(\left(A\land \neg C\right)\to \neg B\right)&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 01. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\left(A\land B\right)\to C&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''Hipótese'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 02.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;A\land \neg C&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp; &amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;'''Hipótese'''
|-
| width=&quot;15px&quot; | 03.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp; &amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;'''2 S'''
|-
| width=&quot;15px&quot; | 04.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg C&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp; &amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;'''2 S'''
|-
| width=&quot;15px&quot; | 05.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \left(A\land B\right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp; &amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;'''1,4 MT'''
|-
| width=&quot;15px&quot; | 06.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg A\lor \neg B&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp; &amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;'''5 DM'''
|-
| width=&quot;15px&quot; | 07.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \neg A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp; &amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;'''3 DN'''
|-
| width=&quot;15px&quot; | 08.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg B&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp; &amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;'''6,7 SD'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 09.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\left(\left(A\land \neg C\right)\to \neg B\right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''2,8 RPC'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 10.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\left(\left(A\land B\right)\to C\right)\to \left(\left(A\land \neg C\right)\to \neg B\right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''1,9 RPC'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 11. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\left(A\land \neg C\right)\to \neg B&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''Hipótese'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 12.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;A\land B&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;
&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;'''Hipótese'''
|-
| 13.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A&lt;/math&gt;
| &amp;nbsp;
| '''12 S'''
|-
| 14.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;B&lt;/math&gt;
| &amp;nbsp;
| '''12 S'''
|-
| 15.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \neg B&lt;/math&gt;
| &amp;nbsp;
| '''14 DN'''
|-
| 16.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \left(A\land \neg C\right)&lt;/math&gt;
| &amp;nbsp;
| '''11,15 MT'''
|-
| 17.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg A\lor \neg \neg C&lt;/math&gt;
| &amp;nbsp;
| '''16 DM'''
|-
| 18.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \neg A&lt;/math&gt;
| &amp;nbsp;
| '''13 DN'''
|-
| 19.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \neg C&lt;/math&gt;
| &amp;nbsp;
| '''17,18 SD'''
|-
| 20.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt; C&lt;/math&gt;
| &amp;nbsp;
| '''19 DN'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 21.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\left(A\land B\right)\to C&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''12,20 RPC'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 22.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\left(\left(A\land \neg C\right)\to \neg B\right)\to \left(\left(A\land B\right)\to C\right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''9,15 RPC'''
|-
|23.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\left(\left(A\land B\right)\to C\right)\leftrightarrow \left(\left(A\land \neg C\right)\to \neg B\right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''10,22 CB'''
|}
|}

====8====
&lt;math&gt;\vdash\left(A \to \left(B \to C \right) \right)\to \left(\left( A \to B \right) \to \left(A \to C \right)\right)&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 1. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;A \to \left(B \to C \right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''Hipótese'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 2.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;A\to B&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp; &amp;nbsp; &amp;nbsp; '''Hipótese'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 3. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;A&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
| '''Hipótese'''
|-
| 4.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;B\to C&lt;/math&gt;
| &amp;nbsp;
| '''1,3 MP'''
|-
| 5.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;B&lt;/math&gt;
| &amp;nbsp;
| '''2,3 MP'''  
|-
| 6.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;C&lt;/math&gt;
| &amp;nbsp;
| '''4,5 MP'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 7.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A\to C&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''3,6 RPC'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 8.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\left( A \to B \right) \to \left(A \to C \right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''2,7 RPC'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 9.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\left(A \to \left(B \to C \right) \right)\to \left(\left( A \to B \right) \to \left(A \to C \right)\right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''1,8 RPC'''
|}
|}

====9====
&lt;math&gt;\vdash \left( D \to \left(B \to A \right) \right) \to \left( B \to \left( D \to A \right) \right)&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 1. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;D \to \left(B \to A \right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''Hipótese'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 2.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;B&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp; &amp;nbsp; &amp;nbsp; '''Hipótese'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 3. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;D&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
| '''Hipótese'''
|-
| 4.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;B\to A&lt;/math&gt;
| &amp;nbsp;
| '''1,3 MP'''
|-
| 5.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;A&lt;/math&gt;
| &amp;nbsp;
| '''4,2 MP'''  
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 6.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;D\to A&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''3,5 RPC'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 7.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;B\to \left(D\to A\right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''2,6 RPC'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 8.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\left( D \to \left(B \to A \right) \right) \to \left( B \to \left( D \to A \right) \right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''1,7 RPC'''
|}
|}

====10====
&lt;math&gt;\vdash \left(P\to Q\right)\to \left(\left(P\to \neg Q\right)\to \neg P\right)&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 1. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;P\to Q&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''Hipótese'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 2.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;P\to \neg Q&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp; &amp;nbsp; &amp;nbsp; '''Hipótese'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 3. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;P&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
| '''Hipótese'''
|-
| 4.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;Q&lt;/math&gt;
| &amp;nbsp;
| '''1,3 MP'''
|-
| 5.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg Q&lt;/math&gt;
| &amp;nbsp;
| '''2,3 MP'''  
|-
| 6.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;Q\land \neg Q&lt;/math&gt;
| &amp;nbsp;
| '''4,5 C'''  
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 7.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg P&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''3,6 RAA'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 8.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\left(P\to \neg Q\right)\to \neg P&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''2,7 RPC'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 9.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\left(P\to Q\right)\to \left(\left(P\to \neg Q\right)\to \neg P\right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''1,8 RPC'''
|}
|}

====11====
&lt;math&gt;\vdash \left(A\to B\right)\to \left(\left(C\to B\right)\to \left(\left(A \lor C\right)\to B\right)\right)&lt;/math&gt;

{|- cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;
| width=&quot;20px&quot; | &amp;nbsp; ||
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 01. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;A\to B&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''Hipótese'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 02.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;C\to B&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| &amp;nbsp; &amp;nbsp; &amp;nbsp; '''Hipótese'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 03. 
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-bottom:2px solid black;&quot; | &lt;math&gt;\neg B&lt;/math&gt;
| width=&quot;15px&quot; | &amp;nbsp;
| '''Hipótese'''
|-
| 04.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg A&lt;/math&gt;
| &amp;nbsp;
| '''1,3 MT'''
|-
| 05.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg C&lt;/math&gt;
| &amp;nbsp;
| '''2,3 MT'''  
|-
| 06.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg A\land \neg C&lt;/math&gt;
| &amp;nbsp;
| '''4,5 C'''  
|-
| 07.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg \left(A\lor C\right)&lt;/math&gt;
| &amp;nbsp;
| '''6 DM'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 08.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\neg B\to \neg \left(A\lor C\right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''3,7 RPC'''
|- 
| width=&quot;15px&quot; | 09.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\left(A\lor C\right)\to B&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''8 CT'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 10.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\left(C\to B\right)\to \left(\left(A \lor C\right)\to B\right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''2,7 RPC'''
|- style=&quot;height:10px&quot;
|
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
|
|
|
|}
{|- cellspacing=&quot;0&quot; cellpadding=&quot;4&quot; border=&quot;0&quot;
| width=&quot;15px&quot; | 11.
| style=&quot;border-right:2px solid black&quot; | &amp;nbsp;
| &lt;math&gt;\left(A\to B\right)\to \left(\left(C\to B\right)\to \left(\left(A \lor C\right)\to B\right)\right)&lt;/math&gt;
| width=&quot;10px&quot; | &amp;nbsp;
| '''1,10 RPC'''
|}
|}

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