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Theorem u3lemnonb 677
Description: Lemma for Kalmbach implication study.
Assertion
Ref Expression
u3lemnonb ((a ->3 b)' v b') = ((a v b') ^ (a' v b'))

Proof of Theorem u3lemnonb
StepHypRef Expression
1 u3lemab 612 . . . 4 ((a ->3 b) ^ b) = ((a ^ b) v (a' ^ b))
2 ax-a2 31 . . . . 5 ((a ^ b) v (a' ^ b)) = ((a' ^ b) v (a ^ b))
3 anor2 89 . . . . . 6 (a' ^ b) = (a v b')'
4 df-a 40 . . . . . 6 (a ^ b) = (a' v b')'
53, 42or 72 . . . . 5 ((a' ^ b) v (a ^ b)) = ((a v b')' v (a' v b')')
62, 5ax-r2 36 . . . 4 ((a ^ b) v (a' ^ b)) = ((a v b')' v (a' v b')')
71, 6ax-r2 36 . . 3 ((a ->3 b) ^ b) = ((a v b')' v (a' v b')')
8 df-a 40 . . 3 ((a ->3 b) ^ b) = ((a ->3 b)' v b')'
9 oran3 93 . . 3 ((a v b')' v (a' v b')') = ((a v b') ^ (a' v b'))'
107, 8, 93tr2 64 . 2 ((a ->3 b)' v b')' = ((a v b') ^ (a' v b'))'
1110con1 66 1 ((a ->3 b)' v b') = ((a v b') ^ (a' v b'))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->3 wi3 14
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-r3 439
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133
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