Theorem nomcon2 295

Description: Lemma for "Non-Orthomodular Models..." paper.

Assertion

Ref	Expression
nomcon2	(a ==2 b) = (b_|_ ==1 a_|_)

Proof of Theorem nomcon2

Step	Hyp	Ref	Expression
1		ax-a2 30	. . . 4 (a v b_|_) = (b_|_ v a)
2		ax-a1 29	. . . . 5 a = a_|__|_
3	2	lor 66	. . . 4 (b_|_ v a) = (b_|_ v a_|__|_)
4	1, 3	ax-r2 35	. . 3 (a v b_|_) = (b_|_ v a_|__|_)
5		ax-a1 29	. . . 4 b = b_|__|_
6		ancom 68	. . . 4 (a_|_ ^ b_|_) = (b_|_ ^ a_|_)
7	5, 6	2or 67	. . 3 (b v (a_|_ ^ b_|_)) = (b_|__|_ v (b_|_ ^ a_|_))
8	4, 7	2an 72	. 2 ((a v b_|_) ^ (b v (a_|_ ^ b_|_))) = ((b_|_ v a_|__|_) ^ (b_|__|_ v (b_|_ ^ a_|_)))
9		df-id2 50	. 2 (a ==2 b) = ((a v b_|_) ^ (b v (a_|_ ^ b_|_)))
10		df-id1 49	. 2 (b_|_ ==1 a_|_) = ((b_|_ v a_|__|_) ^ (b_|__|_ v (b_|_ ^ a_|_)))
11	8, 9, 10	3tr1 60	1 (a ==2 b) = (b_|_ ==1 a_|_)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ==1 wid1 19   ==2 wid2 20

This theorem is referenced by:  nomcon3 296  nom52 325

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-id1 49  df-id2 50

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