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Theorem i3abs3 506

Description: Antecedent absorption.

**Assertion**
Ref	Expression
i3abs3	((a →₃b) →₃((a →₃b) →₃a)) = ((a →₃b) →₃a)

**Proof of Theorem i3abs3**
Step	Hyp	Ref	Expression
1		df-t 40	. . . . . . . 8 1 = (a ∪ a^⊥ )
2	1	lan 70	. . . . . . 7 ((a →₃b)^⊥ ∩ 1) = ((a →₃b)^⊥ ∩ (a ∪ a^⊥ ))
3		an1 98	. . . . . . 7 ((a →₃b)^⊥ ∩ 1) = (a →₃b)^⊥
4		comi31 490	. . . . . . . . . 10 a C (a →₃b)
5	4	comcom 435	. . . . . . . . 9 (a →₃b) C a
6	5	comcom3 436	. . . . . . . 8 (a →₃b)^⊥ C a
7	5	comcom4 437	. . . . . . . 8 (a →₃b)^⊥ C a^⊥
8	6, 7	fh1 451	. . . . . . 7 ((a →₃b)^⊥ ∩ (a ∪ a^⊥ )) = (((a →₃b)^⊥ ∩ a) ∪ ((a →₃b)^⊥ ∩ a^⊥ ))
9	2, 3, 8	3tr2 61	. . . . . 6 (a →₃b)^⊥ = (((a →₃b)^⊥ ∩ a) ∪ ((a →₃b)^⊥ ∩ a^⊥ ))
10	9	ax-r1 34	. . . . 5 (((a →₃b)^⊥ ∩ a) ∪ ((a →₃b)^⊥ ∩ a^⊥ )) = (a →₃b)^⊥
11		comid 179	. . . . . . . 8 (a →₃b) C (a →₃b)
12	11	comcom2 175	. . . . . . 7 (a →₃b) C (a →₃b)^⊥
13	12, 5	fh1 451	. . . . . 6 ((a →₃b) ∩ ((a →₃b)^⊥ ∪ a)) = (((a →₃b) ∩ (a →₃b)^⊥ ) ∪ ((a →₃b) ∩ a))
14		ax-a2 30	. . . . . . 7 (0 ∪ ((a →₃b) ∩ a)) = (((a →₃b) ∩ a) ∪ 0)
15		dff 93	. . . . . . . 8 0 = ((a →₃b) ∩ (a →₃b)^⊥ )
16	15	ax-r5 37	. . . . . . 7 (0 ∪ ((a →₃b) ∩ a)) = (((a →₃b) ∩ (a →₃b)^⊥ ) ∪ ((a →₃b) ∩ a))
17		or0 94	. . . . . . 7 (((a →₃b) ∩ a) ∪ 0) = ((a →₃b) ∩ a)
18	14, 16, 17	3tr2 61	. . . . . 6 (((a →₃b) ∩ (a →₃b)^⊥ ) ∪ ((a →₃b) ∩ a)) = ((a →₃b) ∩ a)
19	13, 18	ax-r2 35	. . . . 5 ((a →₃b) ∩ ((a →₃b)^⊥ ∪ a)) = ((a →₃b) ∩ a)
20	10, 19	2or 67	. . . 4 ((((a →₃b)^⊥ ∩ a) ∪ ((a →₃b)^⊥ ∩ a^⊥ )) ∪ ((a →₃b) ∩ ((a →₃b)^⊥ ∪ a))) = ((a →₃b)^⊥ ∪ ((a →₃b) ∩ a))
21	12, 5	fh4 454	. . . . 5 ((a →₃b)^⊥ ∪ ((a →₃b) ∩ a)) = (((a →₃b)^⊥ ∪ (a →₃b)) ∩ ((a →₃b)^⊥ ∪ a))
22		ax-a2 30	. . . . . . . . 9 ((a →₃b)^⊥ ∪ (a →₃b)) = ((a →₃b) ∪ (a →₃b)^⊥ )
23		df-t 40	. . . . . . . . . 10 1 = ((a →₃b) ∪ (a →₃b)^⊥ )
24	23	ax-r1 34	. . . . . . . . 9 ((a →₃b) ∪ (a →₃b)^⊥ ) = 1
25	22, 24	ax-r2 35	. . . . . . . 8 ((a →₃b)^⊥ ∪ (a →₃b)) = 1
26	25	ran 71	. . . . . . 7 (((a →₃b)^⊥ ∪ (a →₃b)) ∩ ((a →₃b)^⊥ ∪ a)) = (1 ∩ ((a →₃b)^⊥ ∪ a))
27		ancom 68	. . . . . . 7 (1 ∩ ((a →₃b)^⊥ ∪ a)) = (((a →₃b)^⊥ ∪ a) ∩ 1)
28	26, 27	ax-r2 35	. . . . . 6 (((a →₃b)^⊥ ∪ (a →₃b)) ∩ ((a →₃b)^⊥ ∪ a)) = (((a →₃b)^⊥ ∪ a) ∩ 1)
29		an1 98	. . . . . 6 (((a →₃b)^⊥ ∪ a) ∩ 1) = ((a →₃b)^⊥ ∪ a)
30	28, 29	ax-r2 35	. . . . 5 (((a →₃b)^⊥ ∪ (a →₃b)) ∩ ((a →₃b)^⊥ ∪ a)) = ((a →₃b)^⊥ ∪ a)
31	21, 30	ax-r2 35	. . . 4 ((a →₃b)^⊥ ∪ ((a →₃b) ∩ a)) = ((a →₃b)^⊥ ∪ a)
32	20, 31	ax-r2 35	. . 3 ((((a →₃b)^⊥ ∩ a) ∪ ((a →₃b)^⊥ ∩ a^⊥ )) ∪ ((a →₃b) ∩ ((a →₃b)^⊥ ∪ a))) = ((a →₃b)^⊥ ∪ a)
33	32	ax-r1 34	. 2 ((a →₃b)^⊥ ∪ a) = ((((a →₃b)^⊥ ∩ a) ∪ ((a →₃b)^⊥ ∩ a^⊥ )) ∪ ((a →₃b) ∩ ((a →₃b)^⊥ ∪ a)))
34		lem4 493	. 2 ((a →₃b) →₃((a →₃b) →₃a)) = ((a →₃b)^⊥ ∪ a)
35		df-i3 45	. 2 ((a →₃b) →₃a) = ((((a →₃b)^⊥ ∩ a) ∪ ((a →₃b)^⊥ ∩ a^⊥ )) ∪ ((a →₃b) ∩ ((a →₃b)^⊥ ∪ a)))
36	33, 34, 35	3tr1 60	1 ((a →₃b) →₃((a →₃b) →₃a)) = ((a →₃b) →₃a)

Colors of variables: term

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

This theorem is referenced by: i3th7 531 i3th8 532

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 ax-r3 421

This theorem depends on definitions: df-b 38 df-a 39 df-t 40 df-f 41 df-i3 45 df-le1 122 df-le2 123 df-c1 124 df-c2 125

metamath.org