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Theorem vneulem5 1135
 Description: Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 (Contributed by NM, 15-Mar-2010.) (Revised by NM, 31-Mar-2011.)
Assertion
Ref Expression
vneulem5 (((xy) ∪ u) ∩ ((xy) ∪ w)) = ((xy) ∪ (((xy) ∪ u) ∩ w))

Proof of Theorem vneulem5
StepHypRef Expression
1 ancom 74 . 2 (((xy) ∪ u) ∩ ((xy) ∪ w)) = (((xy) ∪ w) ∩ ((xy) ∪ u))
2 ml 1123 . . 3 ((xy) ∪ (w ∩ ((xy) ∪ u))) = (((xy) ∪ w) ∩ ((xy) ∪ u))
32cm 61 . 2 (((xy) ∪ w) ∩ ((xy) ∪ u)) = ((xy) ∪ (w ∩ ((xy) ∪ u)))
4 ancom 74 . . 3 (w ∩ ((xy) ∪ u)) = (((xy) ∪ u) ∩ w)
54lor 70 . 2 ((xy) ∪ (w ∩ ((xy) ∪ u))) = ((xy) ∪ (((xy) ∪ u) ∩ w))
61, 3, 53tr 65 1 (((xy) ∪ u) ∩ ((xy) ∪ w)) = ((xy) ∪ (((xy) ∪ u) ∩ w))
 Colors of variables: term Syntax hints:   = wb 1   ∪ 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 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:  vneulem6  1136  vneulem9  1139
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