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Theorem vneulem9 1139
 Description: Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 (Contributed by NM, 31-Mar-2011.)
Hypothesis
Ref Expression
vneulem6.1 ((ab) ∩ (cd)) = 0
Assertion
Ref Expression
vneulem9 (((ab) ∪ d) ∩ ((ab) ∪ c)) = ((cd) ∪ (ab))

Proof of Theorem vneulem9
StepHypRef Expression
1 ancom 74 . . 3 (((ab) ∪ d) ∩ ((ab) ∪ c)) = (((ab) ∪ c) ∩ ((ab) ∪ d))
2 vneulem5 1135 . . 3 (((ab) ∪ c) ∩ ((ab) ∪ d)) = ((ab) ∪ (((ab) ∪ c) ∩ d))
31, 2ax-r2 36 . 2 (((ab) ∪ d) ∩ ((ab) ∪ c)) = ((ab) ∪ (((ab) ∪ c) ∩ d))
4 orcom 73 . 2 ((ab) ∪ (((ab) ∪ c) ∩ d)) = ((((ab) ∪ c) ∩ d) ∪ (ab))
5 vneulem6.1 . . . 4 ((ab) ∩ (cd)) = 0
65vneulem4 1134 . . 3 (((ab) ∪ c) ∩ d) = (cd)
76ror 71 . 2 ((((ab) ∪ c) ∩ d) ∪ (ab)) = ((cd) ∪ (ab))
83, 4, 73tr 65 1 (((ab) ∪ d) ∩ ((ab) ∪ c)) = ((cd) ∪ (ab))
 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 1122 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:  vneulem11  1141  vneulem16  1146
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