Theorem nomcon0 293

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

Assertion

Ref	Expression
nomcon0	(a ==0 b) = (b_|_ ==0 a_|_)

Proof of Theorem nomcon0

Step	Hyp	Ref	Expression
1		ax-a2 30	. . . 4 (a_|_ v b) = (b v a_|_)
2		ax-a1 29	. . . . 5 b = b_|__|_
3	2	ax-r5 37	. . . 4 (b v a_|_) = (b_|__|_ v a_|_)
4	1, 3	ax-r2 35	. . 3 (a_|_ v b) = (b_|__|_ v a_|_)
5		ax-a2 30	. . . 4 (b_|_ v a) = (a v b_|_)
6		ax-a1 29	. . . . 5 a = a_|__|_
7	6	ax-r5 37	. . . 4 (a v b_|_) = (a_|__|_ v b_|_)
8	5, 7	ax-r2 35	. . 3 (b_|_ v a) = (a_|__|_ v b_|_)
9	4, 8	2an 72	. 2 ((a_|_ v b) ^ (b_|_ v a)) = ((b_|__|_ v a_|_) ^ (a_|__|_ v b_|_))
10		df-id0 48	. 2 (a ==0 b) = ((a_|_ v b) ^ (b_|_ v a))
11		df-id0 48	. 2 (b_|_ ==0 a_|_) = ((b_|__|_ v a_|_) ^ (a_|__|_ v b_|_))
12	9, 10, 11	3tr1 60	1 (a ==0 b) = (b_|_ ==0 a_|_)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ==0 wid0 18

This theorem is referenced by:  nom50 323

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-id0 48

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