Theorem ska8 228

Description: Soundness theorem for Kalmbach's quantum propositional logic axiom KA8.

Assertion

Ref	Expression
ska8	((a⊥ ∩ a) ≡ ((a⊥ ∩ a) ∩ b)) = 1

Proof of Theorem ska8

Step	Hyp	Ref	Expression
1		an0 100	. . . . 5 (b ∩ 0) = 0
2	1	ax-r1 34	. . . 4 0 = (b ∩ 0)
3		ancom 68	. . . 4 (b ∩ 0) = (0 ∩ b)
4	2, 3	ax-r2 35	. . 3 0 = (0 ∩ b)
5		dff 93	. . . 4 0 = (a ∩ a⊥ )
6		ancom 68	. . . 4 (a ∩ a⊥ ) = (a⊥ ∩ a)
7	5, 6	ax-r2 35	. . 3 0 = (a⊥ ∩ a)
8	7	ran 71	. . 3 (0 ∩ b) = ((a⊥ ∩ a) ∩ b)
9	4, 7, 8	3tr2 61	. 2 (a⊥ ∩ a) = ((a⊥ ∩ a) ∩ b)
10	9	bi1 110	1 ((a⊥ ∩ a) ≡ ((a⊥ ∩ a) ∩ b)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∩ wa 7  1wt 9  0wf 10

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  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

metamath.org