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Theorem oagen1 1014
Description: "Generalized" OA.
Hypothesis
Ref Expression
oagen1.1 d =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
Assertion
Ref Expression
oagen1 ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
Proof of Theorem oagen1
StepHypRef Expression
1 oagen1.1 . . . . . . 7 d =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
2 df-i0 43 . . . . . . 7 ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) = ((b v c)' v ((a ->2 b) ^ (a ->2 c)))
31, 2lbtr 139 . . . . . 6 d =< ((b v c)' v ((a ->2 b) ^ (a ->2 c)))
43leror 152 . . . . 5 (d v ((a ->2 b) ^ (a ->2 c))) =< (((b v c)' v ((a ->2 b) ^ (a ->2 c))) v ((a ->2 b) ^ (a ->2 c)))
5 ax-a3 32 . . . . . 6 (((b v c)' v ((a ->2 b) ^ (a ->2 c))) v ((a ->2 b) ^ (a ->2 c))) = ((b v c)' v (((a ->2 b) ^ (a ->2 c)) v ((a ->2 b) ^ (a ->2 c))))
6 oridm 110 . . . . . . 7 (((a ->2 b) ^ (a ->2 c)) v ((a ->2 b) ^ (a ->2 c))) = ((a ->2 b) ^ (a ->2 c))
76lor 70 . . . . . 6 ((b v c)' v (((a ->2 b) ^ (a ->2 c)) v ((a ->2 b) ^ (a ->2 c)))) = ((b v c)' v ((a ->2 b) ^ (a ->2 c)))
85, 7ax-r2 36 . . . . 5 (((b v c)' v ((a ->2 b) ^ (a ->2 c))) v ((a ->2 b) ^ (a ->2 c))) = ((b v c)' v ((a ->2 b) ^ (a ->2 c)))
94, 8lbtr 139 . . . 4 (d v ((a ->2 b) ^ (a ->2 c))) =< ((b v c)' v ((a ->2 b) ^ (a ->2 c)))
109lelan 167 . . 3 ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
11 oath1 1004 . . 3 ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
1210, 11lbtr 139 . 2 ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 b) ^ (a ->2 c))
13 lea 160 . . 3 ((a ->2 b) ^ (a ->2 c)) =< (a ->2 b)
14 leor 159 . . 3 ((a ->2 b) ^ (a ->2 c)) =< (d v ((a ->2 b) ^ (a ->2 c)))
1513, 14ler2an 173 . 2 ((a ->2 b) ^ (a ->2 c)) =< ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c))))
1612, 15lebi 145 1 ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   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:  oagen1b  1015  oadist  1019  oadistb  1020
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