pg63_44_AE1MFC_bryan_kok

pg63_44_AE1MFC_bryan_kok

proof

# Comments like these represent my informal thoughts...
1     p => q                                          Hypothesis         %  ok                        
2     r | s                                           Hypothesis         %  ok                        
3     ~s => ~t                                        Hypothesis         %  ok                        
4     ~q | s                                          Hypothesis         %  ok                        
5     ~s                                              Hypothesis         %  ok                        
6     ~p & r => u                                     Hypothesis         %  ok                        
7     w | t                                           Hypothesis         %  ok

# From lines d and e, ~q | s and ~s implies ~q.
8     [ ~q                                            Assumption         %  ok                        
9       ~q ]                                          Hyp 8              %  ok                        
10    [ s                                             Assumption         %  ok                        
11      F                                             ImpE 5,10          %  ok                        
12      ~q ]                                          FalseE 11          %  ok  

# Using modus tollens on line a.
13    ~q                                              OrE 8-9,10-12,4    %  ok 
14    [ p                                             Assumption         %  ok                        
15      q                                             ImpE 1,14          %  ok                        
16      F ]                                           ImpE 13,15         %  ok                        
17    ~p                                              ImpI 14-16         %  ok   

# From lines b and e, r | s and ~s implies r.
18    [ r                                             Assumption         %  ok                        
19      r ]                                           Hyp 18             %  ok                        
20    [ s                                             Assumption         %  ok                        
21      F                                             ImpE 5,20          %  ok                        
22      r ]                                           FalseE 21          %  ok                        
23    r                                               OrE 18-19,20-22,2  %  ok 

# From line f, ~p & r -> u.
24    ~p & r                                          AndI 17,23         %  ok                        
25    u                                               ImpE 6,24          %  ok 

# From lines c and e, ~s -> ~t.
26    ~t                                              ImpE 3,5           %  ok  

# From line g, w | t and ~t implies w.
27    [ w                                             Assumption         %  ok                        
28      w ]                                           Hyp 27             %  ok                        
29    [ t                                             Assumption         %  ok                        
30      F                                             ImpE 26,29         %  ok                        
31      w ]                                           FalseE 30          %  ok                        
32    w                                               OrE 27-28,29-31,7  %  ok                        
33    u & w                                           AndI 25,32         %  ok