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Theorem u3lem3 751
Description: Lemma for unified implication study.
Assertion
Ref Expression
u3lem3 (a ->3 (b ->3 a)) = (a v ((a' ^ b) v (a' ^ b')))
Proof of Theorem u3lem3
StepHypRef Expression
1 df-i3 46 . 2 (a ->3 (b ->3 a)) = (((a' ^ (b ->3 a)) v (a' ^ (b ->3 a)')) v (a ^ (a' v (b ->3 a))))
2 ancom 74 . . . . . . 7 (a' ^ (b ->3 a)) = ((b ->3 a) ^ a')
3 u3lemanb 617 . . . . . . 7 ((b ->3 a) ^ a') = (b' ^ a')
42, 3ax-r2 36 . . . . . 6 (a' ^ (b ->3 a)) = (b' ^ a')
5 ancom 74 . . . . . . 7 (a' ^ (b ->3 a)') = ((b ->3 a)' ^ a')
6 u3lemnanb 657 . . . . . . 7 ((b ->3 a)' ^ a') = (b ^ a')
75, 6ax-r2 36 . . . . . 6 (a' ^ (b ->3 a)') = (b ^ a')
84, 72or 72 . . . . 5 ((a' ^ (b ->3 a)) v (a' ^ (b ->3 a)')) = ((b' ^ a') v (b ^ a'))
9 ancom 74 . . . . . . 7 (b' ^ a') = (a' ^ b')
10 ancom 74 . . . . . . 7 (b ^ a') = (a' ^ b)
119, 102or 72 . . . . . 6 ((b' ^ a') v (b ^ a')) = ((a' ^ b') v (a' ^ b))
12 ax-a2 31 . . . . . 6 ((a' ^ b') v (a' ^ b)) = ((a' ^ b) v (a' ^ b'))
1311, 12ax-r2 36 . . . . 5 ((b' ^ a') v (b ^ a')) = ((a' ^ b) v (a' ^ b'))
148, 13ax-r2 36 . . . 4 ((a' ^ (b ->3 a)) v (a' ^ (b ->3 a)')) = ((a' ^ b) v (a' ^ b'))
15 ax-a2 31 . . . . . . 7 (a' v (b ->3 a)) = ((b ->3 a) v a')
16 u3lemonb 637 . . . . . . 7 ((b ->3 a) v a') = 1
1715, 16ax-r2 36 . . . . . 6 (a' v (b ->3 a)) = 1
1817lan 77 . . . . 5 (a ^ (a' v (b ->3 a))) = (a ^ 1)
19 an1 106 . . . . 5 (a ^ 1) = a
2018, 19ax-r2 36 . . . 4 (a ^ (a' v (b ->3 a))) = a
2114, 202or 72 . . 3 (((a' ^ (b ->3 a)) v (a' ^ (b ->3 a)')) v (a ^ (a' v (b ->3 a)))) = (((a' ^ b) v (a' ^ b')) v a)
22 ax-a2 31 . . 3 (((a' ^ b) v (a' ^ b')) v a) = (a v ((a' ^ b) v (a' ^ b')))
2321, 22ax-r2 36 . 2 (((a' ^ (b ->3 a)) v (a' ^ (b ->3 a)')) v (a ^ (a' v (b ->3 a)))) = (a v ((a' ^ b) v (a' ^ b')))
241, 23ax-r2 36 1 (a ->3 (b ->3 a)) = (a v ((a' ^ b) v (a' ^ b')))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->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
This theorem is referenced by:  u3lem3n  754  u3lem14a  791
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