Theorem vneulem14 1144

Description: Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96. (Contributed by NM, 31-Mar-2011.)

Hypothesis

Ref	Expression
vneulem13.1	((a ∪ b) ∩ (c ∪ d)) = 0

Assertion

Ref	Expression
vneulem14	(((c ∩ d) ∪ (a ∪ b)) ∩ ((c ∪ d) ∪ (a ∩ b))) = ((c ∩ d) ∪ (a ∩ b))

Proof of Theorem vneulem14

Step	Hyp	Ref	Expression
1		vneulem12 1142	. 2 (((c ∩ d) ∪ (a ∪ b)) ∩ ((c ∪ d) ∪ (a ∩ b))) = ((c ∩ d) ∪ ((a ∪ b) ∩ ((c ∪ d) ∪ (a ∩ b))))
2		vneulem13.1	. . 3 ((a ∪ b) ∩ (c ∪ d)) = 0
3	2	vneulem13 1143	. 2 ((c ∩ d) ∪ ((a ∪ b) ∩ ((c ∪ d) ∪ (a ∩ b)))) = ((c ∩ d) ∪ (a ∩ b))
4	1, 3	tr 62	1 (((c ∩ d) ∪ (a ∪ b)) ∩ ((c ∪ d) ∪ (a ∩ b))) = ((c ∩ d) ∪ (a ∩ b))

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:  vneulem16  1146