Theorem fh3r 457

Description: Foulis-Holland Theorem.

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 453	. 2 (a ∪ (b ∩ c)) = ((a ∪ b) ∩ (a ∪ c))
4		ax-a2 30	. 2 ((b ∩ c) ∪ a) = (a ∪ (b ∩ c))
5		ax-a2 30	. . 3 (b ∪ a) = (a ∪ b)
6		ax-a2 30	. . 3 (c ∪ a) = (a ∪ c)
7	5, 6	2an 72	. 2 ((b ∪ a) ∩ (c ∪ a)) = ((a ∪ b) ∩ (a ∪ c))
8	3, 4, 7	3tr1 60	1 ((b ∩ c) ∪ a) = ((b ∪ a) ∩ (c ∪ a))

Syntax hints:   = wb 1   C wc 3   ∪ wo 6   ∩ wa 7

This theorem is referenced by:  fh3rc 463  ud1lem1 542  ud4lem2 564  ud4lem3b 566  ud5lem3 576  u2lembi 703  u4lem6 750  u1lem11 762  u3lem13b 772  mhlem 858  gomaex3lem2 895  gomaex3lem3 896

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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