Theorem fh3rc 463

Description: Foulis-Holland Theorem.

Hypotheses

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

Assertion

Ref	Expression
fh3rc	((c ∩ b) ∪ a) = ((c ∪ a) ∩ (b ∪ a))

Proof of Theorem fh3rc

Step	Hyp	Ref	Expression
1		fh.1	. . 3 a C b
2		fh.2	. . 3 a C c
3	1, 2	fh3r 457	. 2 ((b ∩ c) ∪ a) = ((b ∪ a) ∩ (c ∪ a))
4		ancom 68	. . 3 (c ∩ b) = (b ∩ c)
5	4	ax-r5 37	. 2 ((c ∩ b) ∪ a) = ((b ∩ c) ∪ a)
6		ancom 68	. 2 ((c ∪ a) ∩ (b ∪ a)) = ((b ∪ a) ∩ (c ∪ a))
7	3, 5, 6	3tr1 60	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 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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