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Theorem conb 122
Description: Contraposition law.
Assertion
Ref Expression
conb (a == b) = (a' == b')
Proof of Theorem conb
StepHypRef Expression
1 ax-a2 31 . . 3 ((a ^ b) v (a' ^ b')) = ((a' ^ b') v (a ^ b))
2 ax-a1 30 . . . . 5 a = a''
3 ax-a1 30 . . . . 5 b = b''
42, 32an 79 . . . 4 (a ^ b) = (a'' ^ b'')
54lor 70 . . 3 ((a' ^ b') v (a ^ b)) = ((a' ^ b') v (a'' ^ b''))
61, 5ax-r2 36 . 2 ((a ^ b) v (a' ^ b')) = ((a' ^ b') v (a'' ^ b''))
7 dfb 94 . 2 (a == b) = ((a ^ b) v (a' ^ b'))
8 dfb 94 . 2 (a' == b') = ((a' ^ b') v (a'' ^ b''))
96, 7, 83tr1 63 1 (a == b) = (a' == 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:  di  126  wr4  199  wcon  202  wcon1  207  wcon2  208  wwfh3  218  wwfh4  219  ka4lem  229  ska3  232  nomcon5  306  nom55  336  wom2  434  u3lemax4  796  comanbn  873
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