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Theorem vneulem11 1139
Description: Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96
Hypothesis
Ref Expression
vneulem6.1 ((a v b) ^ (c v d)) = 0
Assertion
Ref Expression
vneulem11 (((b v c) v d) ^ ((a v c) v d)) = ((c v d) v (a ^ b))
Proof of Theorem vneulem11
StepHypRef Expression
1 ax-a3 32 . . . 4 ((b v c) v d) = (b v (c v d))
2 orcom 73 . . . 4 (b v (c v d)) = ((c v d) v b)
31, 2tr 62 . . 3 ((b v c) v d) = ((c v d) v b)
4 ax-a2 31 . . . . 5 (a v c) = (c v a)
54ror 71 . . . 4 ((a v c) v d) = ((c v a) v d)
6 or32 82 . . . 4 ((c v a) v d) = ((c v d) v a)
75, 6tr 62 . . 3 ((a v c) v d) = ((c v d) v a)
83, 72an 79 . 2 (((b v c) v d) ^ ((a v c) v d)) = (((c v d) v b) ^ ((c v d) v a))
9 ancom 74 . . . 4 ((c v d) ^ (a v b)) = ((a v b) ^ (c v d))
10 vneulem6.1 . . . 4 ((a v b) ^ (c v d)) = 0
119, 10tr 62 . . 3 ((c v d) ^ (a v b)) = 0
1211vneulem9 1137 . 2 (((c v d) v b) ^ ((c v d) v a)) = ((a ^ b) v (c v d))
13 orcom 73 . 2 ((a ^ b) v (c v d)) = ((c v d) v (a ^ b))
148, 12, 133tr 65 1 (((b v c) v d) ^ ((a v c) v d)) = ((c v d) v (a ^ b))
Colors of variables: term
Syntax hints:   = wb 1   v wo 6   ^ wa 7  0wf 9
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-ml 1120
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:  vneulem16  1144
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