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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 == ((a ^ b) v (a ^ b'))) = 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) = (((b ^ a) v (b ^ a')) v ((b' ^ a) v (b' ^ a')))
3 df-t 41 . . . . 5 1 = (b v b')
43bi1 118 . . . 4 (1 == (b v b')) = 1
5 wdf-c1.1 . . . . . 6 (a == ((a ^ b) v (a ^ b'))) = 1
65wcomlem 382 . . . . 5 (b == ((b ^ a) v (b ^ a'))) = 1
7 ax-a1 30 . . . . . . . . . . 11 b = b''
87lan 77 . . . . . . . . . 10 (a ^ b) = (a ^ b'')
98ax-r5 38 . . . . . . . . 9 ((a ^ b) v (a ^ b')) = ((a ^ b'') v (a ^ b'))
10 ax-a2 31 . . . . . . . . 9 ((a ^ b'') v (a ^ b')) = ((a ^ b') v (a ^ b''))
119, 10ax-r2 36 . . . . . . . 8 ((a ^ b) v (a ^ b')) = ((a ^ b') v (a ^ b''))
1211bi1 118 . . . . . . 7 (((a ^ b) v (a ^ b')) == ((a ^ b') v (a ^ b''))) = 1
135, 12wr2 371 . . . . . 6 (a == ((a ^ b') v (a ^ b''))) = 1
1413wcomlem 382 . . . . 5 (b' == ((b' ^ a) v (b' ^ a'))) = 1
156, 14w2or 372 . . . 4 ((b v b') == (((b ^ a) v (b ^ a')) v ((b' ^ a) v (b' ^ a')))) = 1
164, 15wr2 371 . . 3 (1 == (((b ^ a) v (b ^ a')) v ((b' ^ a) v (b' ^ a')))) = 1
1716wr3 198 . 2 (((b ^ a) v (b ^ a')) v ((b' ^ a) v (b' ^ a'))) = 1
181, 2, 173tr 65 1 C (a, b) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   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:  wcom0  407  wcom1  408  wlecom  409  wbctr  410  wcbtr  411  wcomcom2  415  wcomcom5  420  wcomdr  421  wcom3i  422  wcom2or  427
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