document.write( "Question 384302: I need a Formal Logical Proof for the following (I know it is valid):\r
\n" ); document.write( "\n" ); document.write( "(P1) ~(A≡B)
\n" ); document.write( "(P2) ~(B≡C) / A≡C\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "*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:\r
\n" ); document.write( "\n" ); document.write( "~ negation
\n" ); document.write( "∨ or
\n" ); document.write( "• and
\n" ); document.write( "≡ if and only if
\n" ); document.write( "⊃ if ... then \n" ); document.write( "
Algebra.Com's Answer #272202 by jim_thompson5910(35256)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "1.    ~(A = B)\r\n" );
document.write( "2.    ~(B = C)                     /A = C\r\n" );
document.write( "--------------------------------------------\r\n" );
document.write( "3.    ~( (A * B) v (~A * ~B) )           1   Material Equivalence           \r\n" );
document.write( "4.    ~(A * B) * ~(~A * ~B)              3   Distribution\r\n" );
document.write( "5.    (~A v ~B) * (~~A v ~~B)            4   De Morgan's Law\r\n" );
document.write( "6.    (A -> ~B) * (~A -> ~~B)            5   Material Implication\r\n" );
document.write( "7.    (A -> ~B) * (~A -> B)              6   Double Negation\r\n" );
document.write( "8.    (~A -> B) * (A -> ~B)              7   Commutation\r\n" );
document.write( "9.    A -> ~B                            7   Simplification\r\n" );
document.write( "10.   ~A -> B                            8   Simplification\r\n" );
document.write( "11.   ~B -> ~~A                          10  Commutation\r\n" );
document.write( "12.   ~B -> A                            11  Double Negation\r\n" );
document.write( "\r\n" );
document.write( "13.   ~( (B * C) v (~B * ~C) )           2   Material Equivalence      \r\n" );
document.write( "14.  ~(B * C) * ~(~B * ~C)               13   Distribution\r\n" );
document.write( "15.   (~B v ~C) * (~~B v ~~C)            14   De Morgan's Law\r\n" );
document.write( "16.   (B -> ~C) * (~B -> ~~C)            15   Material Implication\r\n" );
document.write( "17.   (B -> ~C) * (~B -> C)              16   Double Negation\r\n" );
document.write( "18.   (~B -> C) * (B -> ~C)              17   Commutation\r\n" );
document.write( "19.    B -> ~C                           17   Simplification\r\n" );
document.write( "20.   ~B -> C                            18   Simplification\r\n" );
document.write( "21.   ~~C -> ~B                          19   Commutation\r\n" );
document.write( "22.   C -> ~B                            21   Double Negation\r\n" );
document.write( "\r\n" );
document.write( "23.   A -> C                           9,20   Hypothetical Syllogism\r\n" );
document.write( "24.   C -> A                          22,12   Hypothetical Syllogism        \r\n" );
document.write( "25.   (A -> C) * (C -> A)             23,24   Conjunction\r\n" );
document.write( "26.   A = C                              25   Material Equivalence\r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: There are two identities for material equivalence.
\n" ); document.write( " \n" ); document.write( "
\n" );