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Theorem oadistc 1022
Description: Distributive law.
Hypotheses
Ref Expression
oadistc.1 d =< ((a ->2 b) ^ (a ->2 c))
oadistc.2 ((a ->2 b) ^ ((b v c)' v d)) =< (((a ->2 b) ^ (b v c)') v d)
Assertion
Ref Expression
oadistc ((a ->2 b) ^ ((b v c)' v d)) = (((a ->2 b) ^ (b v c)') v ((a ->2 b) ^ d))
Proof of Theorem oadistc
StepHypRef Expression
1 oadistc.2 . . 3 ((a ->2 b) ^ ((b v c)' v d)) =< (((a ->2 b) ^ (b v c)') v d)
2 oadistc.1 . . . . . . . 8 d =< ((a ->2 b) ^ (a ->2 c))
3 lea 160 . . . . . . . 8 ((a ->2 b) ^ (a ->2 c)) =< (a ->2 b)
42, 3letr 137 . . . . . . 7 d =< (a ->2 b)
54df2le2 136 . . . . . 6 (d ^ (a ->2 b)) = d
65ax-r1 35 . . . . 5 d = (d ^ (a ->2 b))
7 ancom 74 . . . . 5 (d ^ (a ->2 b)) = ((a ->2 b) ^ d)
86, 7ax-r2 36 . . . 4 d = ((a ->2 b) ^ d)
98lor 70 . . 3 (((a ->2 b) ^ (b v c)') v d) = (((a ->2 b) ^ (b v c)') v ((a ->2 b) ^ d))
101, 9lbtr 139 . 2 ((a ->2 b) ^ ((b v c)' v d)) =< (((a ->2 b) ^ (b v c)') v ((a ->2 b) ^ d))
11 ledi 174 . 2 (((a ->2 b) ^ (b v c)') v ((a ->2 b) ^ d)) =< ((a ->2 b) ^ ((b v c)' v d))
1210, 11lebi 145 1 ((a ->2 b) ^ ((b v c)' v d)) = (((a ->2 b) ^ (b v c)') v ((a ->2 b) ^ d))
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->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-le1 130  df-le2 131
This theorem is referenced by: (None)
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