SOLUTION: I need to construct a proof using all of the rules: (ex. commutation, association, material implication, exportation, hypothetical syllogism, DeMorgans, Double Negation, Distributi

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Question 189101: I need to construct a proof using all of the rules: (ex. commutation, association, material implication, exportation, hypothetical syllogism, DeMorgans, Double Negation, Distribution, etc.)
1. (p <--> q) -> s
2. ~(~r -> t)
3. ~q v ~s Therefore: (t v p) -> (~t & ~ q)
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
This is one tricky derivation, so you need to be a little creative with this one.

Note: each premise is used and when a new premise is derived from, I separated it to keep things looking clean. 

1. (p <--> q) -> s
2. ~(~r -> t)
3. ~q v ~s Therefore: (t v p) -> (~t & ~ q)
--------------------------------------------
4.  ~(~~r v t)                               2     Material Implication
5.  ~(r v t)                                 4     Double Negation
6.   ~r & ~t                                 5     De Morgan's Law
7.   ~t & ~r                                 6     Commutation
8.   ~t                                      7     Simplification
---
9.  ~s v ~q                                  3     Commutation
10.  s -> ~q                                 9     Material Implication
---
11. (p <--> q) -> ~q                         1,10  Hypothetical Syllogism 
12. [(p & q) v (~p & ~q) ] -> ~q             11    Material Equivalence
13. ~[(p & q) v (~p & ~q) ] v ~q             12    Material Implication
14. [~(p & q) & ~(~p & ~q) ] v ~q            13    De Morgan's Law
15. ~q v [~(p & q) & ~(~p & ~q) ]            14    Commutation
16. [~q v (~p v ~q)] & [~q v (p v q) ]       15    Distribution
17. ~q v (~q v ~p)                           16    Simplification
18. (~q v ~q) v ~p                           17    Association
19. ~q v ~p                                  18    Tautology
20. ~p v ~q                                  19    Commutation
---
21. ~t & (~p v ~q)                           8,20  Conjunction
22. (~t & ~p) v (~t & ~q)                    21    Distribution
23. ~(t v p) v (~t & ~q)                     22    De Morgan's Law
24. (t v p) -> (~t & ~q)                     23    Material Implication

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