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Theorem vneulem13 1141
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
vneulem13 ((c ^ d) v ((a v b) ^ ((c v d) v (a ^ b)))) = ((c ^ d) v (a ^ b))
Proof of Theorem vneulem13
StepHypRef Expression
1 leao1 162 . . . . . . 7 (a ^ b) =< (a v b)
2 leid 148 . . . . . . 7 (a ^ b) =< (a ^ b)
31, 2ler2an 173 . . . . . 6 (a ^ b) =< ((a v b) ^ (a ^ b))
4 lear 161 . . . . . 6 ((a v b) ^ (a ^ b)) =< (a ^ b)
53, 4lebi 145 . . . . 5 (a ^ b) = ((a v b) ^ (a ^ b))
65lor 70 . . . 4 ((c v d) v (a ^ b)) = ((c v d) v ((a v b) ^ (a ^ b)))
76lan 77 . . 3 ((a v b) ^ ((c v d) v (a ^ b))) = ((a v b) ^ ((c v d) v ((a v b) ^ (a ^ b))))
8 mldual 1122 . . 3 ((a v b) ^ ((c v d) v ((a v b) ^ (a ^ b)))) = (((a v b) ^ (c v d)) v ((a v b) ^ (a ^ b)))
9 vneulem13.1 . . . . 5 ((a v b) ^ (c v d)) = 0
104, 3lebi 145 . . . . 5 ((a v b) ^ (a ^ b)) = (a ^ b)
119, 102or 72 . . . 4 (((a v b) ^ (c v d)) v ((a v b) ^ (a ^ b))) = (0 v (a ^ b))
12 or0r 103 . . . 4 (0 v (a ^ b)) = (a ^ b)
1311, 12tr 62 . . 3 (((a v b) ^ (c v d)) v ((a v b) ^ (a ^ b))) = (a ^ b)
147, 8, 133tr 65 . 2 ((a v b) ^ ((c v d) v (a ^ b))) = (a ^ b)
1514lor 70 1 ((c ^ d) v ((a v b) ^ ((c v d) v (a ^ b)))) = ((c ^ d) v (a ^ 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:  vneulem14  1142
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