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Theorem cmtrcom 190
Description: Commutative law for commutator.
Assertion
Ref Expression
cmtrcom C (a, b) = C (b, a)
Proof of Theorem cmtrcom
StepHypRef Expression
1 ancom 74 . . . . 5 (a ^ b) = (b ^ a)
2 ancom 74 . . . . 5 (a ^ b') = (b' ^ a)
31, 22or 72 . . . 4 ((a ^ b) v (a ^ b')) = ((b ^ a) v (b' ^ a))
4 ancom 74 . . . . 5 (a' ^ b) = (b ^ a')
5 ancom 74 . . . . 5 (a' ^ b') = (b' ^ a')
64, 52or 72 . . . 4 ((a' ^ b) v (a' ^ b')) = ((b ^ a') v (b' ^ a'))
73, 62or 72 . . 3 (((a ^ b) v (a ^ b')) v ((a' ^ b) v (a' ^ b'))) = (((b ^ a) v (b' ^ a)) v ((b ^ a') v (b' ^ a')))
8 or4 84 . . 3 (((b ^ a) v (b' ^ a)) v ((b ^ a') v (b' ^ a'))) = (((b ^ a) v (b ^ a')) v ((b' ^ a) v (b' ^ a')))
97, 8ax-r2 36 . 2 (((a ^ b) v (a ^ b')) v ((a' ^ b) v (a' ^ b'))) = (((b ^ a) v (b ^ a')) v ((b' ^ a) v (b' ^ a')))
10 df-cmtr 134 . 2 C (a, b) = (((a ^ b) v (a ^ b')) v ((a' ^ b) v (a' ^ b')))
11 df-cmtr 134 . 2 C (b, a) = (((b ^ a) v (b ^ a')) v ((b' ^ a) v (b' ^ a')))
129, 10, 113tr1 63 1 C (a, b) = C (b, a)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   C wcmtr 29
This theorem was proved from axioms:  ax-a2 31  ax-a3 32  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-cmtr 134
This theorem is referenced by:  wdf-c1  383  wcomcom  414  3vded3  819
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