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Theorem wcomcom2 415
 Description: Commutation equivalence. Kalmbach 83 p. 23.
Hypothesis
Ref Expression
wcomcom.1 C (a, b) = 1
Assertion
Ref Expression
wcomcom2 C (a, b ) = 1

Proof of Theorem wcomcom2
StepHypRef Expression
1 wcomcom.1 . . . . 5 C (a, b) = 1
21wdf-c2 384 . . . 4 (a ≡ ((ab) ∪ (ab ))) = 1
3 ax-a1 30 . . . . . . 7 b = b
43bi1 118 . . . . . 6 (bb ) = 1
54wlan 370 . . . . 5 ((ab) ≡ (ab )) = 1
65wr5-2v 366 . . . 4 (((ab) ∪ (ab )) ≡ ((ab ) ∪ (ab ))) = 1
72, 6wr2 371 . . 3 (a ≡ ((ab ) ∪ (ab ))) = 1
8 ax-a2 31 . . . 4 ((ab ) ∪ (ab )) = ((ab ) ∪ (ab ))
98bi1 118 . . 3 (((ab ) ∪ (ab )) ≡ ((ab ) ∪ (ab ))) = 1
107, 9wr2 371 . 2 (a ≡ ((ab ) ∪ (ab ))) = 1
1110wdf-c1 383 1 C (a, b ) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ 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:  wcomcom3  416  wcomcom4  417  wfh1  423  wfh2  424  wnbdi  429  ska2  432  ska4  433
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