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Theorem comdr 466
Description: Commutation dual. Kalmbach 83 p. 23.
Hypothesis
Ref Expression
comdr.1 a = ((a v b) ^ (a v b'))
Assertion
Ref Expression
comdr a C b
Proof of Theorem comdr
StepHypRef Expression
1 comdr.1 . . . . 5 a = ((a v b) ^ (a v b'))
2 df-a 40 . . . . . 6 ((a v b) ^ (a v b')) = ((a v b)' v (a v b')')'
3 oran 87 . . . . . . . . 9 (a v b) = (a' ^ b')'
43con2 67 . . . . . . . 8 (a v b)' = (a' ^ b')
5 oran 87 . . . . . . . . 9 (a v b') = (a' ^ b'')'
65con2 67 . . . . . . . 8 (a v b')' = (a' ^ b'')
74, 62or 72 . . . . . . 7 ((a v b)' v (a v b')') = ((a' ^ b') v (a' ^ b''))
87ax-r4 37 . . . . . 6 ((a v b)' v (a v b')')' = ((a' ^ b') v (a' ^ b''))'
92, 8ax-r2 36 . . . . 5 ((a v b) ^ (a v b')) = ((a' ^ b') v (a' ^ b''))'
101, 9ax-r2 36 . . . 4 a = ((a' ^ b') v (a' ^ b''))'
1110con2 67 . . 3 a' = ((a' ^ b') v (a' ^ b''))
1211df-c1 132 . 2 a' C b'
1312comcom5 458 1 a C 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: (None)
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