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Theorem dfnb 95
Description: Negated biconditional.
Assertion
Ref Expression
dfnb (a == b)' = ((a v b) ^ (a' v b'))
Proof of Theorem dfnb
StepHypRef Expression
1 oran 87 . . . 4 ((a ^ b) v (a' ^ b')) = ((a ^ b)' ^ (a' ^ b')')'
21con2 67 . . 3 ((a ^ b) v (a' ^ b'))' = ((a ^ b)' ^ (a' ^ b')')
3 ancom 74 . . 3 ((a ^ b)' ^ (a' ^ b')') = ((a' ^ b')' ^ (a ^ b)')
42, 3ax-r2 36 . 2 ((a ^ b) v (a' ^ b'))' = ((a' ^ b')' ^ (a ^ b)')
5 dfb 94 . . 3 (a == b) = ((a ^ b) v (a' ^ b'))
65ax-r4 37 . 2 (a == b)' = ((a ^ b) v (a' ^ b'))'
7 oran 87 . . 3 (a v b) = (a' ^ b')'
8 df-a 40 . . . . 5 (a ^ b) = (a' v b')'
98con2 67 . . . 4 (a ^ b)' = (a' v b')
109ax-r1 35 . . 3 (a' v b') = (a ^ b)'
117, 102an 79 . 2 ((a v b) ^ (a' v b')) = ((a' ^ b')' ^ (a ^ b)')
124, 6, 113tr1 63 1 (a == b)' = ((a v b) ^ (a' v b'))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   v wo 6   ^ wa 7
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-b 39  df-a 40
This theorem is referenced by:  wnbdi  429  ska2  432  ska4  433  nbdi  486  test2  803
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