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Theorem wql1 293
Description: The 2nd hypothesis is the first ->1 WQL axiom. We show it implies the WOM law.
Hypotheses
Ref Expression
wql1.1 (a ->1 b) = 1
wql1.2 ((a v c) ->1 (b v c)) = 1
wql1.3 c = b
Assertion
Ref Expression
wql1 (a ->2 b) = 1
Proof of Theorem wql1
StepHypRef Expression
1 df-i2 45 . 2 (a ->2 b) = (b v (a' ^ b'))
2 anor3 90 . . 3 (a' ^ b') = (a v b)'
32lor 70 . 2 (b v (a' ^ b')) = (b v (a v b)')
4 ax-a2 31 . . 3 (b v (a v b)') = ((a v b)' v b)
5 wql1.3 . . . . . . . . 9 c = b
65lor 70 . . . . . . . 8 (b v c) = (b v b)
7 oridm 110 . . . . . . . 8 (b v b) = b
86, 7ax-r2 36 . . . . . . 7 (b v c) = b
98ud1lem0a 255 . . . . . 6 ((a v c) ->1 (b v c)) = ((a v c) ->1 b)
109ax-r1 35 . . . . 5 ((a v c) ->1 b) = ((a v c) ->1 (b v c))
115lor 70 . . . . . 6 (a v c) = (a v b)
1211ud1lem0b 256 . . . . 5 ((a v c) ->1 b) = ((a v b) ->1 b)
13 wql1.2 . . . . 5 ((a v c) ->1 (b v c)) = 1
1410, 12, 133tr2 64 . . . 4 ((a v b) ->1 b) = 1
1514wql1lem 287 . . 3 ((a v b)' v b) = 1
164, 15ax-r2 36 . 2 (b v (a v b)') = 1
171, 3, 163tr 65 1 (a ->2 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-a4 33  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  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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