Theorem nomcon1 294

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

Assertion

Ref	Expression
nomcon1	(a ==1 b) = (b_|_ ==2 a_|_)

Proof of Theorem nomcon1

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		ancom 68	. . . . 5 (a ^ b) = (b ^ a)
6		ax-a1 29	. . . . . 6 b = b_|__|_
7	6, 2	2an 72	. . . . 5 (b ^ a) = (b_|__|_ ^ a_|__|_)
8	5, 7	ax-r2 35	. . . 4 (a ^ b) = (b_|__|_ ^ a_|__|_)
9	8	lor 66	. . 3 (a_|_ v (a ^ b)) = (a_|_ v (b_|__|_ ^ a_|__|_))
10	4, 9	2an 72	. 2 ((a v b_|_) ^ (a_|_ v (a ^ b))) = ((b_|_ v a_|__|_) ^ (a_|_ v (b_|__|_ ^ a_|__|_)))
11		df-id1 49	. 2 (a ==1 b) = ((a v b_|_) ^ (a_|_ v (a ^ b)))
12		df-id2 50	. 2 (b_|_ ==2 a_|_) = ((b_|_ v a_|__|_) ^ (a_|_ v (b_|__|_ ^ a_|__|_)))
13	10, 11, 12	3tr1 60	1 (a ==1 b) = (b_|_ ==2 a_|_)

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

This theorem is referenced by:  nomcon4 297  nom51 324

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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