Theorem fh1rc 479

Description: Foulis-Holland Theorem. (Contributed by NM, 10-Mar-2002.)

Hypotheses

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

Assertion

Ref	Expression
fh1rc	((c ∪ b) ∩ a) = ((c ∩ a) ∪ (b ∩ a))

Proof of Theorem fh1rc

Step	Hyp	Ref	Expression
1		fh.1	. . 3 a C b
2		fh.2	. . 3 a C c
3	1, 2	fh1r 473	. 2 ((b ∪ c) ∩ a) = ((b ∩ a) ∪ (c ∩ a))
4		ax-a2 31	. . 3 (c ∪ b) = (b ∪ c)
5	4	ran 78	. 2 ((c ∪ b) ∩ a) = ((b ∪ c) ∩ a)
6		ax-a2 31	. 2 ((c ∩ a) ∪ (b ∩ a)) = ((b ∩ a) ∪ (c ∩ a))
7	3, 5, 6	3tr1 63	1 ((c ∪ b) ∩ a) = ((c ∩ a) ∪ (b ∩ 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:  mhlemlem1  874