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Theorem wlem1 243
Description: Lemma for 2-variable WOML proof.
Assertion
Ref Expression
wlem1 ((a == b)' v ((a ->1 b) ^ (b ->1 a))) = 1
Proof of Theorem wlem1
StepHypRef Expression
1 le1 146 . 2 ((a == b)' v ((a ->1 b) ^ (b ->1 a))) =< 1
2 df-t 41 . . . 4 1 = ((a == b) v (a == b)')
3 ax-a2 31 . . . 4 ((a == b) v (a == b)') = ((a == b)' v (a == b))
42, 3ax-r2 36 . . 3 1 = ((a == b)' v (a == b))
5 dfb 94 . . . . 5 (a == b) = ((a ^ b) v (a' ^ b'))
6 ledio 176 . . . . . 6 ((a ^ b) v (a' ^ b')) =< (((a ^ b) v a') ^ ((a ^ b) v b'))
7 df-i1 44 . . . . . . . . 9 (a ->1 b) = (a' v (a ^ b))
8 ax-a2 31 . . . . . . . . 9 (a' v (a ^ b)) = ((a ^ b) v a')
97, 8ax-r2 36 . . . . . . . 8 (a ->1 b) = ((a ^ b) v a')
10 df-i1 44 . . . . . . . . 9 (b ->1 a) = (b' v (b ^ a))
11 ax-a2 31 . . . . . . . . . 10 (b' v (b ^ a)) = ((b ^ a) v b')
12 ancom 74 . . . . . . . . . . 11 (b ^ a) = (a ^ b)
1312ax-r5 38 . . . . . . . . . 10 ((b ^ a) v b') = ((a ^ b) v b')
1411, 13ax-r2 36 . . . . . . . . 9 (b' v (b ^ a)) = ((a ^ b) v b')
1510, 14ax-r2 36 . . . . . . . 8 (b ->1 a) = ((a ^ b) v b')
169, 152an 79 . . . . . . 7 ((a ->1 b) ^ (b ->1 a)) = (((a ^ b) v a') ^ ((a ^ b) v b'))
1716ax-r1 35 . . . . . 6 (((a ^ b) v a') ^ ((a ^ b) v b')) = ((a ->1 b) ^ (b ->1 a))
186, 17lbtr 139 . . . . 5 ((a ^ b) v (a' ^ b')) =< ((a ->1 b) ^ (b ->1 a))
195, 18bltr 138 . . . 4 (a == b) =< ((a ->1 b) ^ (b ->1 a))
2019lelor 166 . . 3 ((a == b)' v (a == b)) =< ((a == b)' v ((a ->1 b) ^ (b ->1 a)))
214, 20bltr 138 . 2 1 =< ((a == b)' v ((a ->1 b) ^ (b ->1 a)))
221, 21lebi 145 1 ((a == b)' v ((a ->1 b) ^ (b ->1 a))) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   v wo 6   ^ wa 7  1wt 8   ->1 wi1 12
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
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131
This theorem is referenced by:  wr5-2v  366
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