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Theorem u1lem3 749
Description: Lemma for unified implication study.
Assertion
Ref Expression
u1lem3 (a ->1 (b ->1 a)) = (a' v ((a ^ b) v (a ^ b')))
Proof of Theorem u1lem3
StepHypRef Expression
1 df-i1 44 . 2 (a ->1 (b ->1 a)) = (a' v (a ^ (b ->1 a)))
2 ancom 74 . . . . . . . 8 (a ^ b) = (b ^ a)
3 ancom 74 . . . . . . . 8 (a ^ b') = (b' ^ a)
42, 32or 72 . . . . . . 7 ((a ^ b) v (a ^ b')) = ((b ^ a) v (b' ^ a))
5 u1lemab 610 . . . . . . . 8 ((b ->1 a) ^ a) = ((b ^ a) v (b' ^ a))
65ax-r1 35 . . . . . . 7 ((b ^ a) v (b' ^ a)) = ((b ->1 a) ^ a)
74, 6ax-r2 36 . . . . . 6 ((a ^ b) v (a ^ b')) = ((b ->1 a) ^ a)
8 ancom 74 . . . . . 6 ((b ->1 a) ^ a) = (a ^ (b ->1 a))
97, 8ax-r2 36 . . . . 5 ((a ^ b) v (a ^ b')) = (a ^ (b ->1 a))
109ax-r1 35 . . . 4 (a ^ (b ->1 a)) = ((a ^ b) v (a ^ b'))
1110lor 70 . . 3 (a' v (a ^ (b ->1 a))) = (a' v ((a ^ b) v (a ^ b')))
12 id 59 . . 3 (a' v ((a ^ b) v (a ^ b'))) = (a' v ((a ^ b) v (a ^ b')))
1311, 12ax-r2 36 . 2 (a' v (a ^ (b ->1 a))) = (a' v ((a ^ b) v (a ^ b')))
141, 13ax-r2 36 1 (a ->1 (b ->1 a)) = (a' v ((a ^ b) v (a ^ b')))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12
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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  u1lem4  757
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