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Theorem oaidlem2g 932
Description: Lemma for identity-like OA law (generalized).
Hypothesis
Ref Expression
oaidlem2g.1 ((c v (a ^ b))' v (a ->1 b)) = 1
Assertion
Ref Expression
oaidlem2g (a ^ (c v (a ^ b))) =< b

Proof of Theorem oaidlem2g
StepHypRef Expression
1 anidm 111 . . . . . . . . . 10 (a ^ a) = a
21ax-r1 35 . . . . . . . . 9 a = (a ^ a)
32ran 78 . . . . . . . 8 (a ^ b) = ((a ^ a) ^ b)
4 anass 76 . . . . . . . 8 ((a ^ a) ^ b) = (a ^ (a ^ b))
53, 4ax-r2 36 . . . . . . 7 (a ^ b) = (a ^ (a ^ b))
6 leor 159 . . . . . . . 8 (a ^ b) =< (c v (a ^ b))
76lelan 167 . . . . . . 7 (a ^ (a ^ b)) =< (a ^ (c v (a ^ b)))
85, 7bltr 138 . . . . . 6 (a ^ b) =< (a ^ (c v (a ^ b)))
98df-le2 131 . . . . 5 ((a ^ b) v (a ^ (c v (a ^ b)))) = (a ^ (c v (a ^ b)))
10 ax-a3 32 . . . . . 6 (((c v (a ^ b))' v a') v (a ^ b)) = ((c v (a ^ b))' v (a' v (a ^ b)))
11 ax-a2 31 . . . . . . . 8 ((c v (a ^ b))' v a') = (a' v (c v (a ^ b))')
12 oran3 93 . . . . . . . 8 (a' v (c v (a ^ b))') = (a ^ (c v (a ^ b)))'
1311, 12ax-r2 36 . . . . . . 7 ((c v (a ^ b))' v a') = (a ^ (c v (a ^ b)))'
1413ax-r5 38 . . . . . 6 (((c v (a ^ b))' v a') v (a ^ b)) = ((a ^ (c v (a ^ b)))' v (a ^ b))
15 df-i1 44 . . . . . . . . 9 (a ->1 b) = (a' v (a ^ b))
1615lor 70 . . . . . . . 8 ((c v (a ^ b))' v (a ->1 b)) = ((c v (a ^ b))' v (a' v (a ^ b)))
1716ax-r1 35 . . . . . . 7 ((c v (a ^ b))' v (a' v (a ^ b))) = ((c v (a ^ b))' v (a ->1 b))
18 oaidlem2g.1 . . . . . . 7 ((c v (a ^ b))' v (a ->1 b)) = 1
1917, 18ax-r2 36 . . . . . 6 ((c v (a ^ b))' v (a' v (a ^ b))) = 1
2010, 14, 193tr2 64 . . . . 5 ((a ^ (c v (a ^ b)))' v (a ^ b)) = 1
219, 20lem3.1 443 . . . 4 (a ^ b) = (a ^ (c v (a ^ b)))
2221ax-r1 35 . . 3 (a ^ (c v (a ^ b))) = (a ^ b)
2322bile 142 . 2 (a ^ (c v (a ^ b))) =< (a ^ b)
24 lear 161 . 2 (a ^ b) =< b
2523, 24letr 137 1 (a ^ (c v (a ^ b))) =< b
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   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-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-i1 44  df-le1 130  df-le2 131
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