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	(((x ∪ y) ∪ u) ∩ ((x ∪ y) ∪ w)) = ((x ∪ y) ∪ (((x ∪ y) ∪ u) ∩ w))

Proof of Theorem vneulem5

Step	Hyp	Ref	Expression
1		ancom 74	. 2 (((x ∪ y) ∪ u) ∩ ((x ∪ y) ∪ w)) = (((x ∪ y) ∪ w) ∩ ((x ∪ y) ∪ u))
2		ml 1123	. . 3 ((x ∪ y) ∪ (w ∩ ((x ∪ y) ∪ u))) = (((x ∪ y) ∪ w) ∩ ((x ∪ y) ∪ u))
3	2	cm 61	. 2 (((x ∪ y) ∪ w) ∩ ((x ∪ y) ∪ u)) = ((x ∪ y) ∪ (w ∩ ((x ∪ y) ∪ u)))
4		ancom 74	. . 3 (w ∩ ((x ∪ y) ∪ u)) = (((x ∪ y) ∪ u) ∩ w)
5	4	lor 70	. 2 ((x ∪ y) ∪ (w ∩ ((x ∪ y) ∪ u))) = ((x ∪ y) ∪ (((x ∪ y) ∪ u) ∩ w))
6	1, 3, 5	3tr 65	1 (((x ∪ y) ∪ u) ∩ ((x ∪ y) ∪ w)) = ((x ∪ y) ∪ (((x ∪ y) ∪ u) ∩ w))

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