Theorem fh3r 475

Description: Foulis-Holland Theorem.

Hypotheses

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

Assertion

Ref	Expression
fh3r	((b ^ c) v a) = ((b v a) ^ (c v 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 v (b ^ c)) = ((a v b) ^ (a v c))
4		ax-a2 31	. 2 ((b ^ c) v a) = (a v (b ^ c))
5		ax-a2 31	. . 3 (b v a) = (a v b)
6		ax-a2 31	. . 3 (c v a) = (a v c)
7	5, 6	2an 79	. 2 ((b v a) ^ (c v a)) = ((a v b) ^ (a v c))
8	3, 4, 7	3tr1 63	1 ((b ^ c) v a) = ((b v a) ^ (c v a))

Syntax hints:   = wb 1   C wc 3   v wo 6   ^ wa 7

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

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

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