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Theorem wcomd 418
Description: Commutation dual. Kalmbach 83 p. 23.
Hypothesis
Ref Expression
wcomcom.1 C (a, b) = 1
Assertion
Ref Expression
wcomd (a == ((a v b) ^ (a v b'))) = 1
Proof of Theorem wcomd
StepHypRef Expression
1 wcomcom.1 . . . . 5 C (a, b) = 1
21wcomcom4 417 . . . 4 C (a', b') = 1
32wdf-c2 384 . . 3 (a' == ((a' ^ b') v (a' ^ b''))) = 1
43wcon3 209 . 2 (a == ((a' ^ b') v (a' ^ b''))') = 1
5 oran 87 . . . . 5 ((a' ^ b') v (a' ^ b'')) = ((a' ^ b')' ^ (a' ^ b'')')'
65bi1 118 . . . 4 (((a' ^ b') v (a' ^ b'')) == ((a' ^ b')' ^ (a' ^ b'')')') = 1
76wcon2 208 . . 3 (((a' ^ b') v (a' ^ b''))' == ((a' ^ b')' ^ (a' ^ b'')')) = 1
8 oran 87 . . . . . 6 (a v b) = (a' ^ b')'
98bi1 118 . . . . 5 ((a v b) == (a' ^ b')') = 1
10 oran 87 . . . . . 6 (a v b') = (a' ^ b'')'
1110bi1 118 . . . . 5 ((a v b') == (a' ^ b'')') = 1
129, 11w2an 373 . . . 4 (((a v b) ^ (a v b')) == ((a' ^ b')' ^ (a' ^ b'')')) = 1
1312wr1 197 . . 3 (((a' ^ b')' ^ (a' ^ b'')') == ((a v b) ^ (a v b'))) = 1
147, 13wr2 371 . 2 (((a' ^ b') v (a' ^ b''))' == ((a v b) ^ (a v b'))) = 1
154, 14wr2 371 1 (a == ((a v b) ^ (a v 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:  wcom3ii  419
  Copyright terms: Public domain W3C validator