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Theorem wcomcom5 420
Description: Commutation equivalence. Kalmbach 83 p. 23.
Hypothesis
Ref Expression
wcomcom5.1 C (a', b') = 1
Assertion
Ref Expression
wcomcom5 C (a, b) = 1
Proof of Theorem wcomcom5
StepHypRef Expression
1 wcomcom5.1 . . . . 5 C (a', b') = 1
21wcomcom4 417 . . . 4 C (a'', b'') = 1
32wdf-c2 384 . . 3 (a'' == ((a'' ^ b'') v (a'' ^ b'''))) = 1
4 ax-a1 30 . . . 4 a = a''
54bi1 118 . . 3 (a == a'') = 1
6 ax-a1 30 . . . . . 6 b = b''
76bi1 118 . . . . 5 (b == b'') = 1
85, 7w2an 373 . . . 4 ((a ^ b) == (a'' ^ b'')) = 1
9 ax-a1 30 . . . . . 6 b' = b'''
109bi1 118 . . . . 5 (b' == b''') = 1
115, 10w2an 373 . . . 4 ((a ^ b') == (a'' ^ b''')) = 1
128, 11w2or 372 . . 3 (((a ^ b) v (a ^ b')) == ((a'' ^ b'') v (a'' ^ b'''))) = 1
133, 5, 12w3tr1 374 . 2 (a == ((a ^ b) v (a ^ b'))) = 1
1413wdf-c1 383 1 C (a, b) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v 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:  wcomdr  421  wcom2an  428  woml6  436  woml7  437
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