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Theorem vneulem10 1138
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
vneulem10 (((a v b) v c) ^ ((a v c) v d)) = (a v c)
Proof of Theorem vneulem10
StepHypRef Expression
1 ax-a2 31 . . . 4 (a v b) = (b v a)
21ax-r5 38 . . 3 ((a v b) v c) = ((b v a) v c)
3 or32 82 . . 3 ((a v c) v d) = ((a v d) v c)
42, 32an 79 . 2 (((a v b) v c) ^ ((a v c) v d)) = (((b v a) v c) ^ ((a v d) v c))
5 orcom 73 . . . . 5 (b v a) = (a v b)
6 orcom 73 . . . . 5 (d v c) = (c v d)
75, 62an 79 . . . 4 ((b v a) ^ (d v c)) = ((a v b) ^ (c v d))
8 vneulem6.1 . . . 4 ((a v b) ^ (c v d)) = 0
97, 8tr 62 . . 3 ((b v a) ^ (d v c)) = 0
109vneulem8 1136 . 2 (((b v a) v c) ^ ((a v d) v c)) = (a v c)
114, 10tr 62 1 (((a v b) v c) ^ ((a v c) v d)) = (a v c)
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:  vneulem15  1143
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