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Theorem vneulem12 1140
Description: Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96
Assertion
Ref Expression
vneulem12 (((c ^ d) v (a v b)) ^ ((c v d) v (a ^ b))) = ((c ^ d) v ((a v b) ^ ((c v d) v (a ^ b))))
Proof of Theorem vneulem12
StepHypRef Expression
1 ml 1121 . . 3 ((c ^ d) v ((a v b) ^ ((c ^ d) v ((c v d) v (a ^ b))))) = (((c ^ d) v (a v b)) ^ ((c ^ d) v ((c v d) v (a ^ b))))
21cm 61 . 2 (((c ^ d) v (a v b)) ^ ((c ^ d) v ((c v d) v (a ^ b)))) = ((c ^ d) v ((a v b) ^ ((c ^ d) v ((c v d) v (a ^ b)))))
3 orass 75 . . . . 5 (((c ^ d) v (c v d)) v (a ^ b)) = ((c ^ d) v ((c v d) v (a ^ b)))
43cm 61 . . . 4 ((c ^ d) v ((c v d) v (a ^ b))) = (((c ^ d) v (c v d)) v (a ^ b))
5 leao1 162 . . . . . 6 (c ^ d) =< (c v d)
65df-le2 131 . . . . 5 ((c ^ d) v (c v d)) = (c v d)
76ror 71 . . . 4 (((c ^ d) v (c v d)) v (a ^ b)) = ((c v d) v (a ^ b))
84, 7tr 62 . . 3 ((c ^ d) v ((c v d) v (a ^ b))) = ((c v d) v (a ^ b))
98lan 77 . 2 (((c ^ d) v (a v b)) ^ ((c ^ d) v ((c v d) v (a ^ b)))) = (((c ^ d) v (a v b)) ^ ((c v d) v (a ^ b)))
108lan 77 . . 3 ((a v b) ^ ((c ^ d) v ((c v d) v (a ^ b)))) = ((a v b) ^ ((c v d) v (a ^ b)))
1110lor 70 . 2 ((c ^ d) v ((a v b) ^ ((c ^ d) v ((c v d) v (a ^ b))))) = ((c ^ d) v ((a v b) ^ ((c v d) v (a ^ b))))
122, 9, 113tr2 64 1 (((c ^ d) v (a v b)) ^ ((c v d) v (a ^ b))) = ((c ^ d) v ((a v b) ^ ((c v d) v (a ^ b))))
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:  vneulem14  1142
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