Theorem mh2 884

Description: Marsden-Herman distributive law. Corollary 3.3 of Beran, p. 259. (Contributed by NM, 10-Mar-2002.)

Hypotheses

Ref	Expression
marsden.1	a C b
marsden.2	b C c
marsden.3	c C d
marsden.4	d C a

Assertion

Ref	Expression
mh2	((a ∪ b) ∩ (c ∪ d)) = (((a ∩ c) ∪ (a ∩ d)) ∪ ((b ∩ c) ∪ (b ∩ d)))

Proof of Theorem mh2

Step	Hyp	Ref	Expression
1		marsden.1	. 2 a C b
2		marsden.4	. . 3 d C a
3	2	comcom 453	. 2 a C d
4		marsden.2	. . 3 b C c
5	4	comcom 453	. 2 c C b
6		marsden.3	. 2 c C d
7	1, 3, 5, 6	mh 879	1 ((a ∪ b) ∩ (c ∪ d)) = (((a ∩ c) ∪ (a ∩ d)) ∪ ((b ∩ c) ∪ (b ∩ d)))

Syntax hints:   = wb 1   C wc 3   ∪ wo 6   ∩ wa 7

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  mhcor1  888