Question 384302: I need a Formal Logical Proof for the following (I know it is valid):
(P1) ~(A≡B)
(P2) ~(B≡C) / A≡C
*I am aware that there are many different variations on the symbols, etc. used in formal logic, so below are the symbols we use in class:
~ negation
∨ or
• and
≡ if and only if
⊃ if ... then
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
1. ~(A = B)
2. ~(B = C) /A = C
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3. ~( (A * B) v (~A * ~B) ) 1 Material Equivalence
4. ~(A * B) * ~(~A * ~B) 3 Distribution
5. (~A v ~B) * (~~A v ~~B) 4 De Morgan's Law
6. (A -> ~B) * (~A -> ~~B) 5 Material Implication
7. (A -> ~B) * (~A -> B) 6 Double Negation
8. (~A -> B) * (A -> ~B) 7 Commutation
9. A -> ~B 7 Simplification
10. ~A -> B 8 Simplification
11. ~B -> ~~A 10 Commutation
12. ~B -> A 11 Double Negation
13. ~( (B * C) v (~B * ~C) ) 2 Material Equivalence
14. ~(B * C) * ~(~B * ~C) 13 Distribution
15. (~B v ~C) * (~~B v ~~C) 14 De Morgan's Law
16. (B -> ~C) * (~B -> ~~C) 15 Material Implication
17. (B -> ~C) * (~B -> C) 16 Double Negation
18. (~B -> C) * (B -> ~C) 17 Commutation
19. B -> ~C 17 Simplification
20. ~B -> C 18 Simplification
21. ~~C -> ~B 19 Commutation
22. C -> ~B 21 Double Negation
23. A -> C 9,20 Hypothetical Syllogism
24. C -> A 22,12 Hypothetical Syllogism
25. (A -> C) * (C -> A) 23,24 Conjunction
26. A = C 25 Material Equivalence
Note: There are two identities for material equivalence.
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