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Theorem vneulem2 1130
Description: Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96
Assertion
Ref Expression
vneulem2 (((x v y) v u) ^ ((u v w) ^ w)) = ((((x v y) ^ (u v w)) v u) ^ w)
Proof of Theorem vneulem2
StepHypRef Expression
1 anass 76 . . 3 ((((x v y) v u) ^ (u v w)) ^ w) = (((x v y) v u) ^ ((u v w) ^ w))
21cm 61 . 2 (((x v y) v u) ^ ((u v w) ^ w)) = ((((x v y) v u) ^ (u v w)) ^ w)
3 ax-a2 31 . . . . 5 ((x v y) v u) = (u v (x v y))
43ran 78 . . . 4 (((x v y) v u) ^ (u v w)) = ((u v (x v y)) ^ (u v w))
5 ml 1121 . . . . 5 (u v ((x v y) ^ (u v w))) = ((u v (x v y)) ^ (u v w))
65cm 61 . . . 4 ((u v (x v y)) ^ (u v w)) = (u v ((x v y) ^ (u v w)))
7 orcom 73 . . . 4 (u v ((x v y) ^ (u v w))) = (((x v y) ^ (u v w)) v u)
84, 6, 73tr 65 . . 3 (((x v y) v u) ^ (u v w)) = (((x v y) ^ (u v w)) v u)
98ran 78 . 2 ((((x v y) v u) ^ (u v w)) ^ w) = ((((x v y) ^ (u v w)) v u) ^ w)
102, 9tr 62 1 (((x v y) v u) ^ ((u v w) ^ w)) = ((((x v y) ^ (u v w)) v u) ^ w)
Colors of variables: term
Syntax hints:   = wb 1   v wo 6   ^ wa 7
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:  vneulem4  1132
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