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Theorem u3lem3n 754
Description: Lemma for unified implication study.
Assertion
Ref Expression
u3lem3n (a ->3 (b ->3 a))' = (a' ^ ((a v b) ^ (a v b')))

Proof of Theorem u3lem3n
StepHypRef Expression
1 u3lem3 751 . . 3 (a ->3 (b ->3 a)) = (a v ((a' ^ b) v (a' ^ b')))
2 ax-a2 31 . . . . . 6 ((a' ^ b) v (a' ^ b')) = ((a' ^ b') v (a' ^ b))
3 anor3 90 . . . . . . . 8 (a' ^ b') = (a v b)'
4 anor2 89 . . . . . . . 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 oran1 91 . . . 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 ->3 (b ->3 a)) = (a' ^ ((a v b) ^ (a v b')))'
1312con2 67 1 (a ->3 (b ->3 a))' = (a' ^ ((a v b) ^ (a v b')))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->3 wi3 14
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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133
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