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) v a) = ((c v a) ^ (b v 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) v a) = ((b v a) ^ (c v a))
4		ancom 68	. . 3 (c ^ b) = (b ^ c)
5	4	ax-r5 37	. 2 ((c ^ b) v a) = ((b ^ c) v a)
6		ancom 68	. 2 ((c v a) ^ (b v a)) = ((b v a) ^ (c v a))
7	3, 5, 6	3tr1 60	1 ((c ^ b) v a) = ((c v a) ^ (b v a))

Syntax hints:   = wb 1   C wc 3   v 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

metamath.org