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Theorem comd 456
Description: Commutation dual. Kalmbach 83 p. 23.
Hypothesis
Ref Expression
comcom.1 a C b
Assertion
Ref Expression
comd a = ((a v b) ^ (a v b'))
Proof of Theorem comd
StepHypRef Expression
1 comcom.1 . . . . 5 a C b
21comcom4 455 . . . 4 a' C b'
32df-c2 133 . . 3 a' = ((a' ^ b') v (a' ^ b''))
43con3 68 . 2 a = ((a' ^ b') v (a' ^ b''))'
5 oran 87 . . . 4 ((a' ^ b') v (a' ^ b'')) = ((a' ^ b')' ^ (a' ^ b'')')'
65con2 67 . . 3 ((a' ^ b') v (a' ^ b''))' = ((a' ^ b')' ^ (a' ^ b'')')
7 oran 87 . . . . 5 (a v b) = (a' ^ b')'
8 oran 87 . . . . 5 (a v b') = (a' ^ b'')'
97, 82an 79 . . . 4 ((a v b) ^ (a v b')) = ((a' ^ b')' ^ (a' ^ b'')')
109ax-r1 35 . . 3 ((a' ^ b')' ^ (a' ^ b'')') = ((a v b) ^ (a v b'))
116, 10ax-r2 36 . 2 ((a' ^ b') v (a' ^ b''))' = ((a v b) ^ (a v b'))
124, 11ax-r2 36 1 a = ((a v b) ^ (a v b'))
Colors of variables: term
Syntax hints:   = wb 1   C wc 3  'wn 4   v wo 6   ^ wa 7
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  com3ii  457  gsth  489
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