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Theorem wnbdi 429
Description: Negated biconditional (distributive form)
Assertion
Ref Expression
wnbdi ((a == b)' == (((a v b) ^ a') v ((a v b) ^ b'))) = 1
Proof of Theorem wnbdi
StepHypRef Expression
1 dfnb 95 . . 3 (a == b)' = ((a v b) ^ (a' v b'))
21bi1 118 . 2 ((a == b)' == ((a v b) ^ (a' v b'))) = 1
3 wcomorr 412 . . . . 5 C (a, (a v b)) = 1
43wcomcom 414 . . . 4 C ((a v b), a) = 1
54wcomcom2 415 . . 3 C ((a v b), a') = 1
6 wcomorr 412 . . . . . 6 C (b, (b v a)) = 1
7 ax-a2 31 . . . . . . 7 (b v a) = (a v b)
87bi1 118 . . . . . 6 ((b v a) == (a v b)) = 1
96, 8wcbtr 411 . . . . 5 C (b, (a v b)) = 1
109wcomcom 414 . . . 4 C ((a v b), b) = 1
1110wcomcom2 415 . . 3 C ((a v b), b') = 1
125, 11wfh1 423 . 2 (((a v b) ^ (a' v b')) == (((a v b) ^ a') v ((a v b) ^ b'))) = 1
132, 12wr2 371 1 ((a == b)' == (((a v b) ^ a') v ((a v b) ^ b'))) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   v wo 6   ^ wa 7  1wt 8
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-wom 361
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131  df-cmtr 134
This theorem is referenced by: (None)
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