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Theorem oalem2 1006
Description: Lemma.
Assertion
Ref Expression
oalem2 ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))) = (a ->2 b)
Proof of Theorem oalem2
StepHypRef Expression
1 ax-a2 31 . . . . . . 7 (b v c) = (c v b)
21ax-r4 37 . . . . . 6 (b v c)' = (c v b)'
3 ancom 74 . . . . . 6 ((a ->2 b) ^ (a ->2 c)) = ((a ->2 c) ^ (a ->2 b))
42, 32or 72 . . . . 5 ((b v c)' v ((a ->2 b) ^ (a ->2 c))) = ((c v b)' v ((a ->2 c) ^ (a ->2 b)))
54lan 77 . . . 4 ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 c) ^ ((c v b)' v ((a ->2 c) ^ (a ->2 b))))
6 oath1 1004 . . . 4 ((a ->2 c) ^ ((c v b)' v ((a ->2 c) ^ (a ->2 b)))) = ((a ->2 c) ^ (a ->2 b))
75, 6ax-r2 36 . . 3 ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 c) ^ (a ->2 b))
87lor 70 . 2 ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))) = ((a ->2 b) v ((a ->2 c) ^ (a ->2 b)))
9 ancom 74 . . 3 ((a ->2 c) ^ (a ->2 b)) = ((a ->2 b) ^ (a ->2 c))
109lor 70 . 2 ((a ->2 b) v ((a ->2 c) ^ (a ->2 b))) = ((a ->2 b) v ((a ->2 b) ^ (a ->2 c)))
11 orabs 120 . 2 ((a ->2 b) v ((a ->2 b) ^ (a ->2 c))) = (a ->2 b)
128, 10, 113tr 65 1 ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))) = (a ->2 b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->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  ax-3oa 998
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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