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Theorem ud2lem1 563
Description: Lemma for unified disjunction.
Assertion
Ref Expression
ud2lem1 ((a ->2 b) ->2 (b ->2 a)) = (a v (a' ^ b'))
Proof of Theorem ud2lem1
StepHypRef Expression
1 df-i2 45 . 2 ((a ->2 b) ->2 (b ->2 a)) = ((b ->2 a) v ((a ->2 b)' ^ (b ->2 a)'))
2 df-i2 45 . . . 4 (b ->2 a) = (a v (b' ^ a'))
3 ud2lem0c 278 . . . . 5 (a ->2 b)' = (b' ^ (a v b))
4 ud2lem0c 278 . . . . 5 (b ->2 a)' = (a' ^ (b v a))
53, 42an 79 . . . 4 ((a ->2 b)' ^ (b ->2 a)') = ((b' ^ (a v b)) ^ (a' ^ (b v a)))
62, 52or 72 . . 3 ((b ->2 a) v ((a ->2 b)' ^ (b ->2 a)')) = ((a v (b' ^ a')) v ((b' ^ (a v b)) ^ (a' ^ (b v a))))
7 ancom 74 . . . . . 6 (b' ^ a') = (a' ^ b')
87lor 70 . . . . 5 (a v (b' ^ a')) = (a v (a' ^ b'))
9 dff 101 . . . . . . . 8 0 = ((b' ^ a') ^ (b' ^ a')')
10 oran 87 . . . . . . . . . 10 (b v a) = (b' ^ a')'
1110ax-r1 35 . . . . . . . . 9 (b' ^ a')' = (b v a)
1211lan 77 . . . . . . . 8 ((b' ^ a') ^ (b' ^ a')') = ((b' ^ a') ^ (b v a))
139, 12ax-r2 36 . . . . . . 7 0 = ((b' ^ a') ^ (b v a))
14 anandir 115 . . . . . . . 8 ((b' ^ a') ^ (b v a)) = ((b' ^ (b v a)) ^ (a' ^ (b v a)))
15 ax-a2 31 . . . . . . . . . 10 (b v a) = (a v b)
1615lan 77 . . . . . . . . 9 (b' ^ (b v a)) = (b' ^ (a v b))
1716ran 78 . . . . . . . 8 ((b' ^ (b v a)) ^ (a' ^ (b v a))) = ((b' ^ (a v b)) ^ (a' ^ (b v a)))
1814, 17ax-r2 36 . . . . . . 7 ((b' ^ a') ^ (b v a)) = ((b' ^ (a v b)) ^ (a' ^ (b v a)))
1913, 18ax-r2 36 . . . . . 6 0 = ((b' ^ (a v b)) ^ (a' ^ (b v a)))
2019ax-r1 35 . . . . 5 ((b' ^ (a v b)) ^ (a' ^ (b v a))) = 0
218, 202or 72 . . . 4 ((a v (b' ^ a')) v ((b' ^ (a v b)) ^ (a' ^ (b v a)))) = ((a v (a' ^ b')) v 0)
22 or0 102 . . . 4 ((a v (a' ^ b')) v 0) = (a v (a' ^ b'))
2321, 22ax-r2 36 . . 3 ((a v (b' ^ a')) v ((b' ^ (a v b)) ^ (a' ^ (b v a)))) = (a v (a' ^ b'))
246, 23ax-r2 36 . 2 ((b ->2 a) v ((a ->2 b)' ^ (b ->2 a)')) = (a v (a' ^ b'))
251, 24ax-r2 36 1 ((a ->2 b) ->2 (b ->2 a)) = (a v (a' ^ b'))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  0wf 9   ->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-t 41  df-f 42  df-i2 45
This theorem is referenced by:  ud2  596
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