SOLUTION: This one is ridiculous, please help me with this proof. Premises: 1. (~B v A) -> (C & D) 2. A -> ~C 3. ~A -> ~B Conclusion: ~(B <-> C)

Question 248179: This one is ridiculous, please help me with this proof.
Premises:
1. (~B v A) -> (C & D)
2. A -> ~C
3. ~A -> ~B
Conclusion:
~(B <-> C)
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

1.  (~B v A) -> (C & D)
2.  A -> ~C
3.  ~A -> ~B               /:.  ~(B <-> C)
---------------------------
4.  ~(~B v A) v (C & D)              1       Material Implication
5.  (~~B & ~A) v (C & D)             4       De Morgan's Law
6.  (B & ~A) v (C & D)               5       Double Negation
7.  [(B & ~A) v C] & [(B & ~A) v D]  6       Distribution
8.  [C v (B & ~A)] & [D v (B & ~A)]  7       Commutation
9.  C v (B & ~A)                     8       Simplification
10. (C v B) & (C v ~A)               9       Distribution
11. C v B                            10      Simplification
12. B v C                            11      Commutation
13. ~~B v C                          12      Double Negation
14. ~B -> C                          13      Material Implication
15. B -> A                           3       Transposition
16. B -> ~C                          15,2    Hypothetical Syllogism
17. (B -> ~C) & (~B -> C)            16,14   Conjunction
18. (~B v ~C) & (~~B v C)            17      Material Implication
19. (~B v ~C) & (~~B v ~~C)          18      Double Negation
20. ~(B & C) & ~(~B & ~C)            19      De Morgan's Law
21. ~[(B & C) v (~B & ~C)]           20      De Morgan's Law
22. ~(B <-> C)                       21      Material Equivalence

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