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Theorem wql2lem4 291
Description: Lemma for ->2 WQL axiom.
Hypotheses
Ref Expression
wql2lem4.1 (((a ^ b') v (a ^ b)) ->2 (a' v (a ^ b))) = 1
wql2lem4.2 ((a ->1 b) v (a ^ b')) = 1
Assertion
Ref Expression
wql2lem4 (a ->1 b) = 1

Proof of Theorem wql2lem4
StepHypRef Expression
1 df-i1 44 . 2 (a ->1 b) = (a' v (a ^ b))
2 id 59 . 2 (a' v (a ^ b)) = (a' v (a ^ b))
3 ax-a2 31 . . . 4 ((a ^ b') v (a' v (a ^ b))) = ((a' v (a ^ b)) v (a ^ b'))
41ax-r5 38 . . . . 5 ((a ->1 b) v (a ^ b')) = ((a' v (a ^ b)) v (a ^ b'))
54ax-r1 35 . . . 4 ((a' v (a ^ b)) v (a ^ b')) = ((a ->1 b) v (a ^ b'))
6 wql2lem4.2 . . . 4 ((a ->1 b) v (a ^ b')) = 1
73, 5, 63tr 65 . . 3 ((a ^ b') v (a' v (a ^ b))) = 1
8 wql2lem4.1 . . . 4 (((a ^ b') v (a ^ b)) ->2 (a' v (a ^ b))) = 1
98wql2lem2 289 . . 3 (((a ^ b') v (a' v (a ^ b)))' v (a' v (a ^ b))) = 1
107, 9skr0 242 . 2 (a' v (a ^ b)) = 1
111, 2, 103tr 65 1 (a ->1 b) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->1 wi1 12   ->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-i1 44  df-i2 45
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