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Theorem vneulem15 1143
Description: Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96
Hypothesis
Ref Expression
vneulem13.1 ((a v b) ^ (c v d)) = 0
Assertion
Ref Expression
vneulem15 ((a v c) ^ (b v d)) = ((((a v b) v c) ^ ((a v c) v d)) ^ (((a v b) v d) ^ ((b v c) v d)))
Proof of Theorem vneulem15
StepHypRef Expression
1 vneulem13.1 . . . 4 ((a v b) ^ (c v d)) = 0
21vneulem10 1138 . . 3 (((a v b) v c) ^ ((a v c) v d)) = (a v c)
31vneulem8 1136 . . 3 (((a v b) v d) ^ ((b v c) v d)) = (b v d)
42, 32an 79 . 2 ((((a v b) v c) ^ ((a v c) v d)) ^ (((a v b) v d) ^ ((b v c) v d))) = ((a v c) ^ (b v d))
54cm 61 1 ((a v c) ^ (b v d)) = ((((a v b) v c) ^ ((a v c) v d)) ^ (((a v b) v d) ^ ((b v c) v d)))
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:  vneulem  1145
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