Theorem skr0 234

Description: Soundness theorem for Kalmbach's quantum propositional logic axiom KR0.

Hypotheses

Ref	Expression
skr0.1	a = 1
skr0.2	(a_|_ v b) = 1

Assertion

Ref	Expression
skr0	b = 1

Proof of Theorem skr0

Step	Hyp	Ref	Expression
1		ax-a2 30	. . 3 (b v 0) = (0 v b)
2		or0 94	. . . 4 (b v 0) = b
3	2	ax-r1 34	. . 3 b = (b v 0)
4		skr0.1	. . . . . 6 a = 1
5	4	ax-r4 36	. . . . 5 a_|_ = 1_|_
6		df-f 41	. . . . . 6 0 = 1_|_
7	6	ax-r1 34	. . . . 5 1_|_ = 0
8	5, 7	ax-r2 35	. . . 4 a_|_ = 0
9	8	ax-r5 37	. . 3 (a_|_ v b) = (0 v b)
10	1, 3, 9	3tr1 60	. 2 b = (a_|_ v b)
11		skr0.2	. 2 (a_|_ v b) = 1
12	10, 11	ax-r2 35	1 b = 1

Syntax hints:   = wb 1  _|_wn 4   v wo 6  1wt 9  0wf 10

This theorem is referenced by:  skmp3 237  wql2lem3 282  wql2lem4 283  wr5-2v 348

This theorem was proved from axioms:  ax-a2 30  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-t 40  df-f 41

metamath.org