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Theorem u2lem7n 775
Description: Lemma for unified implication study.
Assertion
Ref Expression
u2lem7n (a ->2 (a' ->2 b))' = (((a v b) ^ (a' v b)) ^ b')
Proof of Theorem u2lem7n
StepHypRef Expression
1 u2lem7 773 . . 3 (a ->2 (a' ->2 b)) = (((a ^ b') v (a' ^ b')) v b)
2 ax-a2 31 . . . . . . 7 ((a ^ b') v (a' ^ b')) = ((a' ^ b') v (a ^ b'))
3 anor3 90 . . . . . . . 8 (a' ^ b') = (a v b)'
4 anor1 88 . . . . . . . 8 (a ^ b') = (a' v b)'
53, 42or 72 . . . . . . 7 ((a' ^ b') v (a ^ b')) = ((a v b)' v (a' v b)')
62, 5ax-r2 36 . . . . . 6 ((a ^ b') v (a' ^ b')) = ((a v b)' v (a' v b)')
7 oran3 93 . . . . . 6 ((a v b)' v (a' v b)') = ((a v b) ^ (a' v b))'
86, 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)
10 oran2 92 . . . 4 (((a v b) ^ (a' v b))' v b) = (((a v b) ^ (a' v b)) ^ b')'
119, 10ax-r2 36 . . 3 (((a ^ b') v (a' ^ b')) v b) = (((a v b) ^ (a' v b)) ^ b')'
121, 11ax-r2 36 . 2 (a ->2 (a' ->2 b)) = (((a v b) ^ (a' v b)) ^ b')'
1312con2 67 1 (a ->2 (a' ->2 b))' = (((a v b) ^ (a' v b)) ^ 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:  u2lem8  782
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