Theorem fh4rc 482

Description: Foulis-Holland Theorem. (Contributed by NM, 20-Sep-1998.)

Hypotheses

Ref	Expression
fh.1	a C b
fh.2	a C c

Assertion

Ref	Expression
fh4rc	((c ∩ a) ∪ b) = ((c ∪ b) ∩ (a ∪ b))

Proof of Theorem fh4rc

Step	Hyp	Ref	Expression
1		fh.1	. . 3 a C b
2		fh.2	. . 3 a C c
3	1, 2	fh4r 476	. 2 ((a ∩ c) ∪ b) = ((a ∪ b) ∩ (c ∪ b))
4		ancom 74	. . 3 (c ∩ a) = (a ∩ c)
5	4	ax-r5 38	. 2 ((c ∩ a) ∪ b) = ((a ∩ c) ∪ b)
6		ancom 74	. 2 ((c ∪ b) ∩ (a ∪ b)) = ((a ∪ b) ∩ (c ∪ b))
7	3, 5, 6	3tr1 63	1 ((c ∩ a) ∪ b) = ((c ∪ b) ∩ (a ∪ b))

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:  gsth  489  orbi  842  negantlem2  849