Theorem i3id 243

Description: Identity for Kalmbach implication.

Assertion

Ref	Expression
i3id	(a →3 a) = 1

Proof of Theorem i3id

Step	Hyp	Ref	Expression
1		ancom 68	. . . . . . . 8 (a⊥ ∩ a) = (a ∩ a⊥ )
2		dff 93	. . . . . . . . 9 0 = (a ∩ a⊥ )
3	2	ax-r1 34	. . . . . . . 8 (a ∩ a⊥ ) = 0
4	1, 3	ax-r2 35	. . . . . . 7 (a⊥ ∩ a) = 0
5		anidm 103	. . . . . . 7 (a⊥ ∩ a⊥ ) = a⊥
6	4, 5	2or 67	. . . . . 6 ((a⊥ ∩ a) ∪ (a⊥ ∩ a⊥ )) = (0 ∪ a⊥ )
7		ax-a2 30	. . . . . 6 (0 ∪ a⊥ ) = (a⊥ ∪ 0)
8	6, 7	ax-r2 35	. . . . 5 ((a⊥ ∩ a) ∪ (a⊥ ∩ a⊥ )) = (a⊥ ∪ 0)
9		or0 94	. . . . 5 (a⊥ ∪ 0) = a⊥
10	8, 9	ax-r2 35	. . . 4 ((a⊥ ∩ a) ∪ (a⊥ ∩ a⊥ )) = a⊥
11		ax-a2 30	. . . . . . 7 (a⊥ ∪ a) = (a ∪ a⊥ )
12		df-t 40	. . . . . . . 8 1 = (a ∪ a⊥ )
13	12	ax-r1 34	. . . . . . 7 (a ∪ a⊥ ) = 1
14	11, 13	ax-r2 35	. . . . . 6 (a⊥ ∪ a) = 1
15	14	lan 70	. . . . 5 (a ∩ (a⊥ ∪ a)) = (a ∩ 1)
16		an1 98	. . . . 5 (a ∩ 1) = a
17	15, 16	ax-r2 35	. . . 4 (a ∩ (a⊥ ∪ a)) = a
18	10, 17	2or 67	. . 3 (((a⊥ ∩ a) ∪ (a⊥ ∩ a⊥ )) ∪ (a ∩ (a⊥ ∪ a))) = (a⊥ ∪ a)
19	18, 11	ax-r2 35	. 2 (((a⊥ ∩ a) ∪ (a⊥ ∩ a⊥ )) ∪ (a ∩ (a⊥ ∪ a))) = (a ∪ a⊥ )
20		df-i3 45	. 2 (a →3 a) = (((a⊥ ∩ a) ∪ (a⊥ ∩ a⊥ )) ∪ (a ∩ (a⊥ ∪ a)))
21	19, 20, 12	3tr1 60	1 (a →3 a) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9  0wf 10   →₃ wi3 15

This theorem is referenced by:  bina1 274  bina2 275  ska14 496  i3orcom 507  i3ancom 508  i3th4 528

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

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i3 45

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