Theorem fh3r 475

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
fh3r	((b ∩ c) ∪ a) = ((b ∪ a) ∩ (c ∪ a))

Proof of Theorem fh3r

Step	Hyp	Ref	Expression
1		fh.1	. . 3 a C b
2		fh.2	. . 3 a C c
3	1, 2	fh3 471	. 2 (a ∪ (b ∩ c)) = ((a ∪ b) ∩ (a ∪ c))
4		ax-a2 31	. 2 ((b ∩ c) ∪ a) = (a ∪ (b ∩ c))
5		ax-a2 31	. . 3 (b ∪ a) = (a ∪ b)
6		ax-a2 31	. . 3 (c ∪ a) = (a ∪ c)
7	5, 6	2an 79	. 2 ((b ∪ a) ∩ (c ∪ a)) = ((a ∪ b) ∩ (a ∪ c))
8	3, 4, 7	3tr1 63	1 ((b ∩ c) ∪ a) = ((b ∪ a) ∩ (c ∪ a))

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:  fh3rc  481  ud1lem1  560  ud4lem2  582  ud4lem3b  584  ud5lem3  594  u2lembi  721  u4lem6  768  u1lem11  780  u3lem13b  790  mhlem  876  gomaex3lem2  915  gomaex3lem3  916