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Theorem oadistc0 1021
Description: Pre-distributive law.
Hypotheses
Ref Expression
oadistc0.1 d =< ((a ->2 b) ^ (a ->2 c))
oadistc0.2 ((a ->2 c) ^ ((a ->2 b) ^ ((b v c)' v d))) =< (((a ->2 b) ^ (b v c)') v d)
Assertion
Ref Expression
oadistc0 ((a ->2 b) ^ ((b v c)' v d)) = (((a ->2 b) ^ (b v c)') v d)
Proof of Theorem oadistc0
StepHypRef Expression
1 ancom 74 . . . . 5 ((a ->2 c) ^ ((a ->2 b) ^ ((b v c)' v d))) = (((a ->2 b) ^ ((b v c)' v d)) ^ (a ->2 c))
2 oadistc0.1 . . . . . . . . 9 d =< ((a ->2 b) ^ (a ->2 c))
32lelor 166 . . . . . . . 8 ((b v c)' v d) =< ((b v c)' v ((a ->2 b) ^ (a ->2 c)))
43lelan 167 . . . . . . 7 ((a ->2 b) ^ ((b v c)' v d)) =< ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
5 oal2 999 . . . . . . 7 ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)
64, 5letr 137 . . . . . 6 ((a ->2 b) ^ ((b v c)' v d)) =< (a ->2 c)
76df2le2 136 . . . . 5 (((a ->2 b) ^ ((b v c)' v d)) ^ (a ->2 c)) = ((a ->2 b) ^ ((b v c)' v d))
81, 7ax-r2 36 . . . 4 ((a ->2 c) ^ ((a ->2 b) ^ ((b v c)' v d))) = ((a ->2 b) ^ ((b v c)' v d))
98ax-r1 35 . . 3 ((a ->2 b) ^ ((b v c)' v d)) = ((a ->2 c) ^ ((a ->2 b) ^ ((b v c)' v d)))
10 oadistc0.2 . . 3 ((a ->2 c) ^ ((a ->2 b) ^ ((b v c)' v d))) =< (((a ->2 b) ^ (b v c)') v d)
119, 10bltr 138 . 2 ((a ->2 b) ^ ((b v c)' v d)) =< (((a ->2 b) ^ (b v c)') v d)
12 ledior 177 . . 3 (((a ->2 b) ^ (b v c)') v d) =< (((a ->2 b) v d) ^ ((b v c)' v d))
13 ax-a2 31 . . . . 5 ((a ->2 b) v d) = (d v (a ->2 b))
14 lea 160 . . . . . . 7 ((a ->2 b) ^ (a ->2 c)) =< (a ->2 b)
152, 14letr 137 . . . . . 6 d =< (a ->2 b)
1615df-le2 131 . . . . 5 (d v (a ->2 b)) = (a ->2 b)
1713, 16ax-r2 36 . . . 4 ((a ->2 b) v d) = (a ->2 b)
1817ran 78 . . 3 (((a ->2 b) v d) ^ ((b v c)' v d)) = ((a ->2 b) ^ ((b v c)' v d))
1912, 18lbtr 139 . 2 (((a ->2 b) ^ (b v c)') v d) =< ((a ->2 b) ^ ((b v c)' v d))
2011, 19lebi 145 1 ((a ->2 b) ^ ((b v c)' v d)) = (((a ->2 b) ^ (b v c)') v 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  ax-3oa 998
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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