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Theorem u5lem3n 756
Description: Lemma for unified implication study.
Assertion
Ref Expression
u5lem3n (a ->5 (b ->5 a))' = (a ^ ((a' v b) ^ (a' v b')))
Proof of Theorem u5lem3n
StepHypRef Expression
1 u5lem3 753 . . 3 (a ->5 (b ->5 a)) = (a' v ((a ^ b) v (a ^ b')))
2 ax-a2 31 . . . . . 6 ((a ^ b) v (a ^ b')) = ((a ^ b') v (a ^ b))
3 anor1 88 . . . . . . . 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'))'
98lor 70 . . . 4 (a' v ((a ^ b) v (a ^ b'))) = (a' v ((a' v b) ^ (a' v b'))')
10 oran3 93 . . . 4 (a' v ((a' v b) ^ (a' v b'))') = (a ^ ((a' v b) ^ (a' v b')))'
119, 10ax-r2 36 . . 3 (a' v ((a ^ b) v (a ^ b'))) = (a ^ ((a' v b) ^ (a' v b')))'
121, 11ax-r2 36 . 2 (a ->5 (b ->5 a)) = (a ^ ((a' v b) ^ (a' v b')))'
1312con2 67 1 (a ->5 (b ->5 a))' = (a ^ ((a' v b) ^ (a' v b')))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->5 wi5 16
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-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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