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Theorem vneulem16 1144
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
vneulem16 ((((a v b) v c) ^ ((a v c) v d)) ^ (((a v b) v d) ^ ((b v c) v d))) = ((a ^ b) v (c ^ d))
Proof of Theorem vneulem16
StepHypRef Expression
1 ancom 74 . 2 ((((a v b) v c) ^ ((a v c) v d)) ^ (((a v b) v d) ^ ((b v c) v d))) = ((((a v b) v d) ^ ((b v c) v d)) ^ (((a v b) v c) ^ ((a v c) v d)))
2 an4 86 . . 3 ((((a v b) v d) ^ ((b v c) v d)) ^ (((a v b) v c) ^ ((a v c) v d))) = ((((a v b) v d) ^ ((a v b) v c)) ^ (((b v c) v d) ^ ((a v c) v d)))
3 vneulem13.1 . . . . 5 ((a v b) ^ (c v d)) = 0
43vneulem9 1137 . . . 4 (((a v b) v d) ^ ((a v b) v c)) = ((c ^ d) v (a v b))
53vneulem11 1139 . . . 4 (((b v c) v d) ^ ((a v c) v d)) = ((c v d) v (a ^ b))
64, 52an 79 . . 3 ((((a v b) v d) ^ ((a v b) v c)) ^ (((b v c) v d) ^ ((a v c) v d))) = (((c ^ d) v (a v b)) ^ ((c v d) v (a ^ b)))
72, 6tr 62 . 2 ((((a v b) v d) ^ ((b v c) v d)) ^ (((a v b) v c) ^ ((a v c) v d))) = (((c ^ d) v (a v b)) ^ ((c v d) v (a ^ b)))
83vneulem14 1142 . . 3 (((c ^ d) v (a v b)) ^ ((c v d) v (a ^ b))) = ((c ^ d) v (a ^ b))
9 orcom 73 . . 3 ((c ^ d) v (a ^ b)) = ((a ^ b) v (c ^ d))
108, 9tr 62 . 2 (((c ^ d) v (a v b)) ^ ((c v d) v (a ^ b))) = ((a ^ b) v (c ^ d))
111, 7, 103tr 65 1 ((((a v b) v c) ^ ((a v c) v d)) ^ (((a v b) v d) ^ ((b v c) v d))) = ((a ^ b) v (c ^ 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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