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Theorem wql2lem5 292
Description: Lemma for ->2 WQL axiom.
Hypothesis
Ref Expression
wql2lem5.1 (a ->2 b) = 1
Assertion
Ref Expression
wql2lem5 ((b' ^ (a v b)) ->2 a') = 1
Proof of Theorem wql2lem5
StepHypRef Expression
1 anor3 90 . . . 4 ((b' ^ (a v b))' ^ a'') = ((b' ^ (a v b)) v a')'
2 oran3 93 . . . . . 6 ((a ->2 b)' v a') = ((a ->2 b) ^ a)'
3 ud2lem0c 278 . . . . . . 7 (a ->2 b)' = (b' ^ (a v b))
43ax-r5 38 . . . . . 6 ((a ->2 b)' v a') = ((b' ^ (a v b)) v a')
5 wql2lem5.1 . . . . . . . . 9 (a ->2 b) = 1
65ran 78 . . . . . . . 8 ((a ->2 b) ^ a) = (1 ^ a)
7 ancom 74 . . . . . . . 8 (1 ^ a) = (a ^ 1)
8 an1 106 . . . . . . . 8 (a ^ 1) = a
96, 7, 83tr 65 . . . . . . 7 ((a ->2 b) ^ a) = a
109ax-r4 37 . . . . . 6 ((a ->2 b) ^ a)' = a'
112, 4, 103tr2 64 . . . . 5 ((b' ^ (a v b)) v a') = a'
1211ax-r4 37 . . . 4 ((b' ^ (a v b)) v a')' = a''
131, 12ax-r2 36 . . 3 ((b' ^ (a v b))' ^ a'') = a''
1413lor 70 . 2 (a' v ((b' ^ (a v b))' ^ a'')) = (a' v a'')
15 df-i2 45 . 2 ((b' ^ (a v b)) ->2 a') = (a' v ((b' ^ (a v b))' ^ a''))
16 df-t 41 . 2 1 = (a' v a'')
1714, 15, 163tr1 63 1 ((b' ^ (a v b)) ->2 a') = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->2 wi2 13
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  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-i2 45
This theorem is referenced by: (None)
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