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Theorem u4lemnab 653
Description: Lemma for non-tollens implication study.
Assertion
Ref Expression
u4lemnab ((a ->4 b)' ^ b) = (((a v b') ^ (a' v b')) ^ b)
Proof of Theorem u4lemnab
StepHypRef Expression
1 u4lemonb 638 . . . 4 ((a ->4 b) v b') = (((a ^ b) v (a' ^ b)) v b')
2 ax-a2 31 . . . . . 6 ((a ^ b) v (a' ^ b)) = ((a' ^ b) v (a ^ b))
3 anor2 89 . . . . . . . 8 (a' ^ b) = (a v b')'
4 df-a 40 . . . . . . . 8 (a ^ b) = (a' v b')'
53, 42or 72 . . . . . . 7 ((a' ^ b) v (a ^ b)) = ((a v b')' v (a' v b')')
6 oran3 93 . . . . . . 7 ((a v b')' v (a' v b')') = ((a v b') ^ (a' v b'))'
75, 6ax-r2 36 . . . . . 6 ((a' ^ b) v (a ^ b)) = ((a v b') ^ (a' v b'))'
82, 7ax-r2 36 . . . . 5 ((a ^ b) v (a' ^ b)) = ((a v b') ^ (a' v b'))'
98ax-r5 38 . . . 4 (((a ^ b) v (a' ^ b)) v b') = (((a v b') ^ (a' v b'))' v b')
101, 9ax-r2 36 . . 3 ((a ->4 b) v b') = (((a v b') ^ (a' v b'))' v b')
11 oran1 91 . . 3 ((a ->4 b) v b') = ((a ->4 b)' ^ b)'
12 oran3 93 . . 3 (((a v b') ^ (a' v b'))' v b') = (((a v b') ^ (a' v b')) ^ b)'
1310, 11, 123tr2 64 . 2 ((a ->4 b)' ^ b)' = (((a v b') ^ (a' v b')) ^ b)'
1413con1 66 1 ((a ->4 b)' ^ b) = (((a v b') ^ (a' v b')) ^ b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->4 wi4 15
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
This theorem depends on definitions:  df-a 40  df-i4 47  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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