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Theorem u3lem14a 791
Description: Lemma for unified implication study.
Assertion
Ref Expression
u3lem14a (a ->3 ((b ->3 a') ->3 b')) = (a ->3 (b ->3 a))
Proof of Theorem u3lem14a
StepHypRef Expression
1 u3lem13b 790 . . 3 ((b ->3 a') ->3 b') = (b ->1 a)
21ud3lem0a 260 . 2 (a ->3 ((b ->3 a') ->3 b')) = (a ->3 (b ->1 a))
3 df-i3 46 . . 3 (a ->3 (b ->1 a)) = (((a' ^ (b ->1 a)) v (a' ^ (b ->1 a)')) v (a ^ (a' v (b ->1 a))))
4 ancom 74 . . . . . . . 8 (a' ^ (b ->1 a)) = ((b ->1 a) ^ a')
5 u1lemanb 615 . . . . . . . 8 ((b ->1 a) ^ a') = (b' ^ a')
64, 5ax-r2 36 . . . . . . 7 (a' ^ (b ->1 a)) = (b' ^ a')
7 ancom 74 . . . . . . . 8 (a' ^ (b ->1 a)') = ((b ->1 a)' ^ a')
8 u1lemnanb 655 . . . . . . . 8 ((b ->1 a)' ^ a') = (b ^ a')
97, 8ax-r2 36 . . . . . . 7 (a' ^ (b ->1 a)') = (b ^ a')
106, 92or 72 . . . . . 6 ((a' ^ (b ->1 a)) v (a' ^ (b ->1 a)')) = ((b' ^ a') v (b ^ a'))
11 ax-a2 31 . . . . . . 7 ((b' ^ a') v (b ^ a')) = ((b ^ a') v (b' ^ a'))
12 ancom 74 . . . . . . . 8 (b ^ a') = (a' ^ b)
13 ancom 74 . . . . . . . 8 (b' ^ a') = (a' ^ b')
1412, 132or 72 . . . . . . 7 ((b ^ a') v (b' ^ a')) = ((a' ^ b) v (a' ^ b'))
1511, 14ax-r2 36 . . . . . 6 ((b' ^ a') v (b ^ a')) = ((a' ^ b) v (a' ^ b'))
1610, 15ax-r2 36 . . . . 5 ((a' ^ (b ->1 a)) v (a' ^ (b ->1 a)')) = ((a' ^ b) v (a' ^ b'))
17 ax-a2 31 . . . . . . . 8 (a' v (b ->1 a)) = ((b ->1 a) v a')
18 u1lemonb 635 . . . . . . . 8 ((b ->1 a) v a') = 1
1917, 18ax-r2 36 . . . . . . 7 (a' v (b ->1 a)) = 1
2019lan 77 . . . . . 6 (a ^ (a' v (b ->1 a))) = (a ^ 1)
21 an1 106 . . . . . 6 (a ^ 1) = a
2220, 21ax-r2 36 . . . . 5 (a ^ (a' v (b ->1 a))) = a
2316, 222or 72 . . . 4 (((a' ^ (b ->1 a)) v (a' ^ (b ->1 a)')) v (a ^ (a' v (b ->1 a)))) = (((a' ^ b) v (a' ^ b')) v a)
24 ax-a2 31 . . . . 5 (((a' ^ b) v (a' ^ b')) v a) = (a v ((a' ^ b) v (a' ^ b')))
25 u3lem3 751 . . . . . . 7 (a ->3 (b ->3 a)) = (a v ((a' ^ b) v (a' ^ b')))
2625ax-r1 35 . . . . . 6 (a v ((a' ^ b) v (a' ^ b'))) = (a ->3 (b ->3 a))
27 id 59 . . . . . 6 (a ->3 (b ->3 a)) = (a ->3 (b ->3 a))
2826, 27ax-r2 36 . . . . 5 (a v ((a' ^ b) v (a' ^ b'))) = (a ->3 (b ->3 a))
2924, 28ax-r2 36 . . . 4 (((a' ^ b) v (a' ^ b')) v a) = (a ->3 (b ->3 a))
3023, 29ax-r2 36 . . 3 (((a' ^ (b ->1 a)) v (a' ^ (b ->1 a)')) v (a ^ (a' v (b ->1 a)))) = (a ->3 (b ->3 a))
313, 30ax-r2 36 . 2 (a ->3 (b ->1 a)) = (a ->3 (b ->3 a))
322, 31ax-r2 36 1 (a ->3 ((b ->3 a') ->3 b')) = (a ->3 (b ->3 a))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->1 wi1 12   ->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-i1 44  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  u3lem14aa  792
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