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Theorem distoah4 943
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
distoah4 (d ^ (a ->2 c)) =< f
Proof of Theorem distoah4
StepHypRef Expression
1 leo 158 . 2 ((a ->2 b) ^ (a ->2 c)) =< (((a ->2 b) ^ (a ->2 c)) v ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))'))
2 distoa.1 . . 3 d = (a ->2 b)
32ran 78 . 2 (d ^ (a ->2 c)) = ((a ->2 b) ^ (a ->2 c))
4 distoa.3 . . 3 f = ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))
5 df-i2 45 . . 3 ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) = (((a ->2 b) ^ (a ->2 c)) v ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))'))
64, 5ax-r2 36 . 2 f = (((a ->2 b) ^ (a ->2 c)) v ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))'))
71, 3, 6le3tr1 140 1 (d ^ (a ->2 c)) =< f
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12   ->2 wi2 13
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-i2 45  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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