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Theorem oadist2b 1008
Description: Distributive inference derived from OA.
Hypothesis
Ref Expression
oadist2b.1 d =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
Assertion
Ref Expression
oadist2b ((a ->2 b) ^ (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))) = (((a ->2 b) ^ d) v ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))))
Proof of Theorem oadist2b
StepHypRef Expression
1 oadist2b.1 . . . . 5 d =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
2 u12lem 771 . . . . . 6 (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) = ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
32ax-r1 35 . . . . 5 ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) = (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))
41, 3lbtr 139 . . . 4 d =< (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))
5 leor 159 . . . 4 ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) =< (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))
64, 5lel2or 170 . . 3 (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) =< (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))
76, 2lbtr 139 . 2 (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
87oadist2a 1007 1 ((a ->2 b) ^ (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))) = (((a ->2 b) ^ d) v ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))))
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2   v wo 6   ^ wa 7   ->0 wi0 11   ->1 wi1 12   ->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: (None)
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