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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 (ab) = (ba)
2 ancom 74 . . . . 5 (ab ) = (ba)
31, 22or 72 . . . 4 ((ab) ∪ (ab )) = ((ba) ∪ (ba))
4 ancom 74 . . . . 5 (ab) = (ba )
5 ancom 74 . . . . 5 (ab ) = (ba )
64, 52or 72 . . . 4 ((ab) ∪ (ab )) = ((ba ) ∪ (ba ))
73, 62or 72 . . 3 (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab ))) = (((ba) ∪ (ba)) ∪ ((ba ) ∪ (ba )))
8 or4 84 . . 3 (((ba) ∪ (ba)) ∪ ((ba ) ∪ (ba ))) = (((ba) ∪ (ba )) ∪ ((ba) ∪ (ba )))
97, 8ax-r2 36 . 2 (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab ))) = (((ba) ∪ (ba )) ∪ ((ba) ∪ (ba )))
10 df-cmtr 134 . 2 C (a, b) = (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab )))
11 df-cmtr 134 . 2 C (b, a) = (((ba) ∪ (ba )) ∪ ((ba) ∪ (ba )))
129, 10, 113tr1 63 1 C (a, b) = C (b, a)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ 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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