Theorem oml5 431

Description: Orthomodular law.

Assertion

Ref	Expression
oml5	((a ^ b) v ((a ^ b)_|_ ^ (b v c))) = (b v c)

Proof of Theorem oml5

Step	Hyp	Ref	Expression
1		oml 427	. . 3 ((a ^ b) v ((a ^ b)_|_ ^ ((a ^ b) v (b v c)))) = ((a ^ b) v (b v c))
2		ax-a3 31	. . . . . 6 ((b v (a ^ b)) v c) = (b v ((a ^ b) v c))
3		ancom 68	. . . . . . . . 9 (a ^ b) = (b ^ a)
4	3	lor 66	. . . . . . . 8 (b v (a ^ b)) = (b v (b ^ a))
5		a5b 112	. . . . . . . 8 (b v (b ^ a)) = b
6	4, 5	ax-r2 35	. . . . . . 7 (b v (a ^ b)) = b
7	6	ax-r5 37	. . . . . 6 ((b v (a ^ b)) v c) = (b v c)
8		or12 73	. . . . . 6 (b v ((a ^ b) v c)) = ((a ^ b) v (b v c))
9	2, 7, 8	3tr2 61	. . . . 5 (b v c) = ((a ^ b) v (b v c))
10	9	lan 70	. . . 4 ((a ^ b)_|_ ^ (b v c)) = ((a ^ b)_|_ ^ ((a ^ b) v (b v c)))
11	10	lor 66	. . 3 ((a ^ b) v ((a ^ b)_|_ ^ (b v c))) = ((a ^ b) v ((a ^ b)_|_ ^ ((a ^ b) v (b v c))))
12	2, 8	ax-r2 35	. . 3 ((b v (a ^ b)) v c) = ((a ^ b) v (b v c))
13	1, 11, 12	3tr1 60	. 2 ((a ^ b) v ((a ^ b)_|_ ^ (b v c))) = ((b v (a ^ b)) v c)
14	13, 7	ax-r2 35	1 ((a ^ b) v ((a ^ b)_|_ ^ (b v c))) = (b v c)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7

This theorem is referenced by:  i3th1 525

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  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

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