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Theorem wdf-c1 383
 Description: Show that commutator is a 'commutes' analogue for ≡ analogue of =.
Hypothesis
Ref Expression
wdf-c1.1 (a ≡ ((ab) ∪ (ab ))) = 1
Assertion
Ref Expression
wdf-c1 C (a, b) = 1

Proof of Theorem wdf-c1
StepHypRef Expression
1 cmtrcom 190 . 2 C (a, b) = C (b, a)
2 df-cmtr 134 . 2 C (b, a) = (((ba) ∪ (ba )) ∪ ((ba) ∪ (ba )))
3 df-t 41 . . . . 5 1 = (bb )
43bi1 118 . . . 4 (1 ≡ (bb )) = 1
5 wdf-c1.1 . . . . . 6 (a ≡ ((ab) ∪ (ab ))) = 1
65wcomlem 382 . . . . 5 (b ≡ ((ba) ∪ (ba ))) = 1
7 ax-a1 30 . . . . . . . . . . 11 b = b
87lan 77 . . . . . . . . . 10 (ab) = (ab )
98ax-r5 38 . . . . . . . . 9 ((ab) ∪ (ab )) = ((ab ) ∪ (ab ))
10 ax-a2 31 . . . . . . . . 9 ((ab ) ∪ (ab )) = ((ab ) ∪ (ab ))
119, 10ax-r2 36 . . . . . . . 8 ((ab) ∪ (ab )) = ((ab ) ∪ (ab ))
1211bi1 118 . . . . . . 7 (((ab) ∪ (ab )) ≡ ((ab ) ∪ (ab ))) = 1
135, 12wr2 371 . . . . . 6 (a ≡ ((ab ) ∪ (ab ))) = 1
1413wcomlem 382 . . . . 5 (b ≡ ((ba) ∪ (ba ))) = 1
156, 14w2or 372 . . . 4 ((bb ) ≡ (((ba) ∪ (ba )) ∪ ((ba) ∪ (ba )))) = 1
164, 15wr2 371 . . 3 (1 ≡ (((ba) ∪ (ba )) ∪ ((ba) ∪ (ba )))) = 1
1716wr3 198 . 2 (((ba) ∪ (ba )) ∪ ((ba) ∪ (ba ))) = 1
181, 2, 173tr 65 1 C (a, b) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   C wcmtr 29 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:  wcom0  407  wcom1  408  wlecom  409  wbctr  410  wcbtr  411  wcomcom2  415  wcomcom5  420  wcomdr  421  wcom3i  422  wcom2or  427
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