Theorem ska13 233

Description: Soundness theorem for Kalmbach's quantum propositional logic axiom KA13.

Assertion

Ref	Expression
ska13	((a == b)_|_ v (a_|_ v b)) = 1

Proof of Theorem ska13

Step	Hyp	Ref	Expression
1		ledio 168	. . . . 5 ((a ^ b) v (a_|_ ^ b_|_)) =< (((a ^ b) v a_|_) ^ ((a ^ b) v b_|_))
2		lea 152	. . . . 5 (((a ^ b) v a_|_) ^ ((a ^ b) v b_|_)) =< ((a ^ b) v a_|_)
3	1, 2	letr 129	. . . 4 ((a ^ b) v (a_|_ ^ b_|_)) =< ((a ^ b) v a_|_)
4		ancom 68	. . . . . 6 (a ^ b) = (b ^ a)
5		lea 152	. . . . . 6 (b ^ a) =< b
6	4, 5	bltr 130	. . . . 5 (a ^ b) =< b
7	6	leror 144	. . . 4 ((a ^ b) v a_|_) =< (b v a_|_)
8	3, 7	letr 129	. . 3 ((a ^ b) v (a_|_ ^ b_|_)) =< (b v a_|_)
9		dfb 86	. . 3 (a == b) = ((a ^ b) v (a_|_ ^ b_|_))
10		ax-a2 30	. . 3 (a_|_ v b) = (b v a_|_)
11	8, 9, 10	le3tr1 132	. 2 (a == b) =< (a_|_ v b)
12	11	sklem 222	1 ((a == b)_|_ v (a_|_ v b)) = 1

Syntax hints:   = wb 1  _|_wn 4   == tb 5   v wo 6   ^ wa 7  1wt 9

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123

metamath.org