SOLUTION: I'm stumped on this problem, could someone please help me construct a proof? 1. (G * P) -> K 2. E -> Z 3. ~P -> ~Z 4. G -> (E v L) conclusion is (G * ~L) -> K

Question 297357: I'm stumped on this problem, could someone please help me construct a proof?
1. (G * P) -> K
2. E -> Z
3. ~P -> ~Z
4. G -> (E v L) conclusion is (G * ~L) -> K
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
It really helps to fully know the 19 rules of inference.

1.  (G * P) -> K
2.  E -> Z
3.  ~P -> ~Z
4.  G ->(E v L)  :. (G * ~L) -> K
--------------------------------
5.  ~~Z -> ~~P                             3         Transposition
6.  Z -> P                                 5         Double Negation
7.  E -> P                                 2,6       Hypothetical Syllogism
8.  (P * G) -> K                           1         Commutation
9.  P -> (G -> K)                          8         Exportation
10. E -> (G -> K)                          7,9       Hypothetical Syllogism
11. ~G v (E v L)                           4         Material Implication
12. (~G v E) v L                           11        Association
13. (E v ~G) v L                           12        Commutation
14. (~~E v ~G) v L                         13        Double Negation
15. ~(~E * G) v L                          14        De Morgan's Law
16. (~E * G) -> L                          15        Material Implication
17. ~E -> (G -> L)                         16        Exportation
18. ~(G -> L) -> ~~E                       17        Transposition
19. ~(G -> L) -> E                         18        Double Negation
20. ~(G -> L) -> (G -> K)                  19,10     Hypothetical Syllogism
21. (G -> L) v (G -> K)                    20        Material Implication
22. (~G v L) v (~G v K)                    21        Material Implication
23. ~G v (L v ~G) v K                      22        Association
24. ~G v (~G v L) v K                      23        Commutation
25. (~G v ~G) v (L v K)                    24        Association
26. ~G v (L v K)                           25        Tautology
27. (~G v L) v K                           26        Association
28. (~G v ~~L) v K                         27        Double Negation
29. ~(G * ~L) v K                          28        De Morgan's Law
30. (G * ~L) -> K                          29        Material Implication

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