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Theorem u2lem7 773
Description: Lemma for unified implication study.
Assertion
Ref Expression
u2lem7 (a ->2 (a' ->2 b)) = (((a ^ b') v (a' ^ b')) v b)
Proof of Theorem u2lem7
StepHypRef Expression
1 df-i2 45 . 2 (a ->2 (a' ->2 b)) = ((a' ->2 b) v (a' ^ (a' ->2 b)'))
2 df-i2 45 . . . . 5 (a' ->2 b) = (b v (a'' ^ b'))
3 ax-a1 30 . . . . . . . 8 a = a''
43ax-r1 35 . . . . . . 7 a'' = a
54ran 78 . . . . . 6 (a'' ^ b') = (a ^ b')
65lor 70 . . . . 5 (b v (a'' ^ b')) = (b v (a ^ b'))
72, 6ax-r2 36 . . . 4 (a' ->2 b) = (b v (a ^ b'))
8 ancom 74 . . . . 5 (a' ^ (a' ->2 b)') = ((a' ->2 b)' ^ a')
9 u2lemnaa 641 . . . . 5 ((a' ->2 b)' ^ a') = (a' ^ b')
108, 9ax-r2 36 . . . 4 (a' ^ (a' ->2 b)') = (a' ^ b')
117, 102or 72 . . 3 ((a' ->2 b) v (a' ^ (a' ->2 b)')) = ((b v (a ^ b')) v (a' ^ b'))
12 ax-a3 32 . . . 4 ((b v (a ^ b')) v (a' ^ b')) = (b v ((a ^ b') v (a' ^ b')))
13 ax-a2 31 . . . 4 (b v ((a ^ b') v (a' ^ b'))) = (((a ^ b') v (a' ^ b')) v b)
1412, 13ax-r2 36 . . 3 ((b v (a ^ b')) v (a' ^ b')) = (((a ^ b') v (a' ^ b')) v b)
1511, 14ax-r2 36 . 2 ((a' ->2 b) v (a' ^ (a' ->2 b)')) = (((a ^ b') v (a' ^ b')) v b)
161, 15ax-r2 36 1 (a ->2 (a' ->2 b)) = (((a ^ b') v (a' ^ b')) v b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->2 wi2 13
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-i2 45  df-le1 130  df-le2 131
This theorem is referenced by:  u2lem7n  775  u2lem8  782
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