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Theorem u3lem8 783
Description: Lemma for unified implication study.
Assertion
Ref Expression
u3lem8 (a' ->3 (a ->3 (a' ->3 b))) = 1
Proof of Theorem u3lem8
StepHypRef Expression
1 comi31 508 . . . 4 a C (a ->3 (a' ->3 b))
21comcom3 454 . . 3 a' C (a ->3 (a' ->3 b))
32u3lemc4 703 . 2 (a' ->3 (a ->3 (a' ->3 b))) = (a'' v (a ->3 (a' ->3 b)))
4 ax-a1 30 . . . . 5 a = a''
54ax-r1 35 . . . 4 a'' = a
6 u3lem7 774 . . . 4 (a ->3 (a' ->3 b)) = (a' v ((a ^ b) v (a ^ b')))
75, 62or 72 . . 3 (a'' v (a ->3 (a' ->3 b))) = (a v (a' v ((a ^ b) v (a ^ b'))))
8 ax-a3 32 . . . . 5 ((a v a') v ((a ^ b) v (a ^ b'))) = (a v (a' v ((a ^ b) v (a ^ b'))))
98ax-r1 35 . . . 4 (a v (a' v ((a ^ b) v (a ^ b')))) = ((a v a') v ((a ^ b) v (a ^ b')))
10 ax-a2 31 . . . . 5 ((a v a') v ((a ^ b) v (a ^ b'))) = (((a ^ b) v (a ^ b')) v (a v a'))
11 df-t 41 . . . . . . . 8 1 = (a v a')
1211ax-r1 35 . . . . . . 7 (a v a') = 1
1312lor 70 . . . . . 6 (((a ^ b) v (a ^ b')) v (a v a')) = (((a ^ b) v (a ^ b')) v 1)
14 or1 104 . . . . . 6 (((a ^ b) v (a ^ b')) v 1) = 1
1513, 14ax-r2 36 . . . . 5 (((a ^ b) v (a ^ b')) v (a v a')) = 1
1610, 15ax-r2 36 . . . 4 ((a v a') v ((a ^ b) v (a ^ b'))) = 1
179, 16ax-r2 36 . . 3 (a v (a' v ((a ^ b) v (a ^ b')))) = 1
187, 17ax-r2 36 . 2 (a'' v (a ->3 (a' ->3 b))) = 1
193, 18ax-r2 36 1 (a' ->3 (a ->3 (a' ->3 b))) = 1
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: (None)
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