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Theorem distoah2 941
Description: Satisfaction of distributive law hypothesis.
Hypotheses
Ref Expression
distoa.1 d = (a ->2 b)
distoa.2 e = ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))
distoa.3 f = ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))
Assertion
Ref Expression
distoah2 e =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
Proof of Theorem distoah2
StepHypRef Expression
1 leo 158 . 2 ((b v c) ->1 ((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))))
2 distoa.2 . . 3 e = ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))
32ax-r1 35 . 2 ((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) = e
4 u12lem 771 . 2 (((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)))
51, 3, 4le3tr2 141 1 e =< ((b v c) ->0 ((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
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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