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Theorem wcomdr 421
Description: Commutation dual. Kalmbach 83 p. 23.
Hypothesis
Ref Expression
wcomdr.1 (a == ((a v b) ^ (a v b'))) = 1
Assertion
Ref Expression
wcomdr C (a, b) = 1
Proof of Theorem wcomdr
StepHypRef Expression
1 wcomdr.1 . . . . 5 (a == ((a v b) ^ (a v b'))) = 1
2 df-a 40 . . . . . . 7 ((a v b) ^ (a v b')) = ((a v b)' v (a v b')')'
32bi1 118 . . . . . 6 (((a v b) ^ (a v b')) == ((a v b)' v (a v b')')') = 1
4 oran 87 . . . . . . . . . 10 (a v b) = (a' ^ b')'
54bi1 118 . . . . . . . . 9 ((a v b) == (a' ^ b')') = 1
65wcon2 208 . . . . . . . 8 ((a v b)' == (a' ^ b')) = 1
7 oran 87 . . . . . . . . . 10 (a v b') = (a' ^ b'')'
87bi1 118 . . . . . . . . 9 ((a v b') == (a' ^ b'')') = 1
98wcon2 208 . . . . . . . 8 ((a v b')' == (a' ^ b'')) = 1
106, 9w2or 372 . . . . . . 7 (((a v b)' v (a v b')') == ((a' ^ b') v (a' ^ b''))) = 1
1110wr4 199 . . . . . 6 (((a v b)' v (a v b')')' == ((a' ^ b') v (a' ^ b''))') = 1
123, 11wr2 371 . . . . 5 (((a v b) ^ (a v b')) == ((a' ^ b') v (a' ^ b''))') = 1
131, 12wr2 371 . . . 4 (a == ((a' ^ b') v (a' ^ b''))') = 1
1413wcon2 208 . . 3 (a' == ((a' ^ b') v (a' ^ b''))) = 1
1514wdf-c1 383 . 2 C (a', b') = 1
1615wcomcom5 420 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: (None)
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