SOLUTION: 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 differen

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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
--------------------------------------------
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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