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Theorem vneulem9 1137
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
vneulem9 (((a v b) v d) ^ ((a v b) v c)) = ((c ^ d) v (a v b))
Proof of Theorem vneulem9
StepHypRef Expression
1 ancom 74 . . 3 (((a v b) v d) ^ ((a v b) v c)) = (((a v b) v c) ^ ((a v b) v d))
2 vneulem5 1133 . . 3 (((a v b) v c) ^ ((a v b) v d)) = ((a v b) v (((a v b) v c) ^ d))
31, 2ax-r2 36 . 2 (((a v b) v d) ^ ((a v b) v c)) = ((a v b) v (((a v b) v c) ^ d))
4 orcom 73 . 2 ((a v b) v (((a v b) v c) ^ d)) = ((((a v b) v c) ^ d) v (a v b))
5 vneulem6.1 . . . 4 ((a v b) ^ (c v d)) = 0
65vneulem4 1132 . . 3 (((a v b) v c) ^ d) = (c ^ d)
76ror 71 . 2 ((((a v b) v c) ^ d) v (a v b)) = ((c ^ d) v (a v b))
83, 4, 73tr 65 1 (((a v b) v d) ^ ((a v b) v c)) = ((c ^ d) v (a v 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:  vneulem11  1139  vneulem16  1144
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