Theorem fh4r 476

Description: Foulis-Holland Theorem. (Contributed by NM, 23-Nov-1997.)

Hypotheses

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

Assertion

Ref	Expression
fh4r	((a ∩ c) ∪ b) = ((a ∪ b) ∩ (c ∪ b))

Proof of Theorem fh4r

Step	Hyp	Ref	Expression
1		fh.1	. . 3 a C b
2		fh.2	. . 3 a C c
3	1, 2	fh4 472	. 2 (b ∪ (a ∩ c)) = ((b ∪ a) ∩ (b ∪ c))
4		ax-a2 31	. 2 ((a ∩ c) ∪ b) = (b ∪ (a ∩ c))
5		ax-a2 31	. . 3 (a ∪ b) = (b ∪ a)
6		ax-a2 31	. . 3 (c ∪ b) = (b ∪ c)
7	5, 6	2an 79	. 2 ((a ∪ b) ∩ (c ∪ b)) = ((b ∪ a) ∩ (b ∪ c))
8	3, 4, 7	3tr1 63	1 ((a ∩ c) ∪ b) = ((a ∪ b) ∩ (c ∪ 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:  fh4rc  482  ud1lem1  560  ud1lem3  562  ud3lem1c  568  ud3lem3  576  ud4lem1c  579  ud4lem3  585  u4lemoa  623  u24lem  770  u3lem10  785  u3lem13a  789  u3lem13b  790  i1abs  801  test  802  test2  803