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Theorem oagen1b 1015
Description: "Generalized" OA.
Hypotheses
Ref Expression
oagen1b.1 d =< (a ->2 b)
oagen1b.2 e =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
Assertion
Ref Expression
oagen1b (d ^ (e v ((a ->2 b) ^ (a ->2 c)))) = (d ^ (a ->2 c))
Proof of Theorem oagen1b
StepHypRef Expression
1 oagen1b.2 . . . 4 e =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
21oagen1 1014 . . 3 ((a ->2 b) ^ (e v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
32lan 77 . 2 (d ^ ((a ->2 b) ^ (e v ((a ->2 b) ^ (a ->2 c))))) = (d ^ ((a ->2 b) ^ (a ->2 c)))
4 anass 76 . . . 4 ((d ^ (a ->2 b)) ^ (e v ((a ->2 b) ^ (a ->2 c)))) = (d ^ ((a ->2 b) ^ (e v ((a ->2 b) ^ (a ->2 c)))))
54ax-r1 35 . . 3 (d ^ ((a ->2 b) ^ (e v ((a ->2 b) ^ (a ->2 c))))) = ((d ^ (a ->2 b)) ^ (e v ((a ->2 b) ^ (a ->2 c))))
6 oagen1b.1 . . . . 5 d =< (a ->2 b)
76df2le2 136 . . . 4 (d ^ (a ->2 b)) = d
87ran 78 . . 3 ((d ^ (a ->2 b)) ^ (e v ((a ->2 b) ^ (a ->2 c)))) = (d ^ (e v ((a ->2 b) ^ (a ->2 c))))
95, 8ax-r2 36 . 2 (d ^ ((a ->2 b) ^ (e v ((a ->2 b) ^ (a ->2 c))))) = (d ^ (e v ((a ->2 b) ^ (a ->2 c))))
10 anass 76 . . . 4 ((d ^ (a ->2 b)) ^ (a ->2 c)) = (d ^ ((a ->2 b) ^ (a ->2 c)))
1110ax-r1 35 . . 3 (d ^ ((a ->2 b) ^ (a ->2 c))) = ((d ^ (a ->2 b)) ^ (a ->2 c))
127ran 78 . . 3 ((d ^ (a ->2 b)) ^ (a ->2 c)) = (d ^ (a ->2 c))
1311, 12ax-r2 36 . 2 (d ^ ((a ->2 b) ^ (a ->2 c))) = (d ^ (a ->2 c))
143, 9, 133tr2 64 1 (d ^ (e v ((a ->2 b) ^ (a ->2 c)))) = (d ^ (a ->2 c))
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2   v wo 6   ^ wa 7   ->0 wi0 11   ->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-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131
This theorem is referenced by:  oadistd  1023
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