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Theorem vneulem15 1143
 Description: Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96
Hypothesis
Ref Expression
vneulem13.1 ((ab) ∩ (cd)) = 0
Assertion
Ref Expression
vneulem15 ((ac) ∩ (bd)) = ((((ab) ∪ c) ∩ ((ac) ∪ d)) ∩ (((ab) ∪ d) ∩ ((bc) ∪ d)))

Proof of Theorem vneulem15
StepHypRef Expression
1 vneulem13.1 . . . 4 ((ab) ∩ (cd)) = 0
21vneulem10 1138 . . 3 (((ab) ∪ c) ∩ ((ac) ∪ d)) = (ac)
31vneulem8 1136 . . 3 (((ab) ∪ d) ∩ ((bc) ∪ d)) = (bd)
42, 32an 79 . 2 ((((ab) ∪ c) ∩ ((ac) ∪ d)) ∩ (((ab) ∪ d) ∩ ((bc) ∪ d))) = ((ac) ∩ (bd))
54cm 61 1 ((ac) ∩ (bd)) = ((((ab) ∪ c) ∩ ((ac) ∪ d)) ∩ (((ab) ∪ d) ∩ ((bc) ∪ d)))
 Colors of variables: term Syntax hints:   = wb 1   ∪ 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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