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Theorem ka4lem 229
Description: Lemma for KA4 soundness (AND version) - uses OL only.
Assertion
Ref Expression
ka4lem ((a ^ b)' v ((a ^ c) == (b ^ c))) = 1
Proof of Theorem ka4lem
StepHypRef Expression
1 df-a 40 . . . 4 (a ^ b) = (a' v b')'
21con2 67 . . 3 (a ^ b)' = (a' v b')
3 df-a 40 . . . . 5 (a ^ c) = (a' v c')'
4 df-a 40 . . . . 5 (b ^ c) = (b' v c')'
53, 42bi 99 . . . 4 ((a ^ c) == (b ^ c)) = ((a' v c')' == (b' v c')')
6 conb 122 . . . . 5 ((a' v c') == (b' v c')) = ((a' v c')' == (b' v c')')
76ax-r1 35 . . . 4 ((a' v c')' == (b' v c')') = ((a' v c') == (b' v c'))
85, 7ax-r2 36 . . 3 ((a ^ c) == (b ^ c)) = ((a' v c') == (b' v c'))
92, 82or 72 . 2 ((a ^ b)' v ((a ^ c) == (b ^ c))) = ((a' v b') v ((a' v c') == (b' v c')))
10 ka4lemo 228 . 2 ((a' v b') v ((a' v c') == (b' v c'))) = 1
119, 10ax-r2 36 1 ((a ^ b)' v ((a ^ c) == (b ^ c))) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   v wo 6   ^ wa 7  1wt 8
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
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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