Theorem i3abs3 524

Description: Antecedent absorption. (Contributed by NM, 19-Nov-1997.)

Assertion

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

Proof of Theorem i3abs3

Step	Hyp	Ref	Expression
1		df-t 41	. . . . . . . 8 1 = (a ∪ a⊥ )
2	1	lan 77	. . . . . . 7 ((a →3 b)⊥ ∩ 1) = ((a →3 b)⊥ ∩ (a ∪ a⊥ ))
3		an1 106	. . . . . . 7 ((a →3 b)⊥ ∩ 1) = (a →3 b)⊥
4		comi31 508	. . . . . . . . . 10 a C (a →3 b)
5	4	comcom 453	. . . . . . . . 9 (a →3 b) C a
6	5	comcom3 454	. . . . . . . 8 (a →3 b)⊥ C a
7	5	comcom4 455	. . . . . . . 8 (a →3 b)⊥ C a⊥
8	6, 7	fh1 469	. . . . . . 7 ((a →3 b)⊥ ∩ (a ∪ a⊥ )) = (((a →3 b)⊥ ∩ a) ∪ ((a →3 b)⊥ ∩ a⊥ ))
9	2, 3, 8	3tr2 64	. . . . . 6 (a →3 b)⊥ = (((a →3 b)⊥ ∩ a) ∪ ((a →3 b)⊥ ∩ a⊥ ))
10	9	ax-r1 35	. . . . 5 (((a →3 b)⊥ ∩ a) ∪ ((a →3 b)⊥ ∩ a⊥ )) = (a →3 b)⊥
11		comid 187	. . . . . . . 8 (a →3 b) C (a →3 b)
12	11	comcom2 183	. . . . . . 7 (a →3 b) C (a →3 b)⊥
13	12, 5	fh1 469	. . . . . 6 ((a →3 b) ∩ ((a →3 b)⊥ ∪ a)) = (((a →3 b) ∩ (a →3 b)⊥ ) ∪ ((a →3 b) ∩ a))
14		ax-a2 31	. . . . . . 7 (0 ∪ ((a →3 b) ∩ a)) = (((a →3 b) ∩ a) ∪ 0)
15		dff 101	. . . . . . . 8 0 = ((a →3 b) ∩ (a →3 b)⊥ )
16	15	ax-r5 38	. . . . . . 7 (0 ∪ ((a →3 b) ∩ a)) = (((a →3 b) ∩ (a →3 b)⊥ ) ∪ ((a →3 b) ∩ a))
17		or0 102	. . . . . . 7 (((a →3 b) ∩ a) ∪ 0) = ((a →3 b) ∩ a)
18	14, 16, 17	3tr2 64	. . . . . 6 (((a →3 b) ∩ (a →3 b)⊥ ) ∪ ((a →3 b) ∩ a)) = ((a →3 b) ∩ a)
19	13, 18	ax-r2 36	. . . . 5 ((a →3 b) ∩ ((a →3 b)⊥ ∪ a)) = ((a →3 b) ∩ a)
20	10, 19	2or 72	. . . 4 ((((a →3 b)⊥ ∩ a) ∪ ((a →3 b)⊥ ∩ a⊥ )) ∪ ((a →3 b) ∩ ((a →3 b)⊥ ∪ a))) = ((a →3 b)⊥ ∪ ((a →3 b) ∩ a))
21	12, 5	fh4 472	. . . . 5 ((a →3 b)⊥ ∪ ((a →3 b) ∩ a)) = (((a →3 b)⊥ ∪ (a →3 b)) ∩ ((a →3 b)⊥ ∪ a))
22		ax-a2 31	. . . . . . . . 9 ((a →3 b)⊥ ∪ (a →3 b)) = ((a →3 b) ∪ (a →3 b)⊥ )
23		df-t 41	. . . . . . . . . 10 1 = ((a →3 b) ∪ (a →3 b)⊥ )
24	23	ax-r1 35	. . . . . . . . 9 ((a →3 b) ∪ (a →3 b)⊥ ) = 1
25	22, 24	ax-r2 36	. . . . . . . 8 ((a →3 b)⊥ ∪ (a →3 b)) = 1
26	25	ran 78	. . . . . . 7 (((a →3 b)⊥ ∪ (a →3 b)) ∩ ((a →3 b)⊥ ∪ a)) = (1 ∩ ((a →3 b)⊥ ∪ a))
27		ancom 74	. . . . . . 7 (1 ∩ ((a →3 b)⊥ ∪ a)) = (((a →3 b)⊥ ∪ a) ∩ 1)
28	26, 27	ax-r2 36	. . . . . 6 (((a →3 b)⊥ ∪ (a →3 b)) ∩ ((a →3 b)⊥ ∪ a)) = (((a →3 b)⊥ ∪ a) ∩ 1)
29		an1 106	. . . . . 6 (((a →3 b)⊥ ∪ a) ∩ 1) = ((a →3 b)⊥ ∪ a)
30	28, 29	ax-r2 36	. . . . 5 (((a →3 b)⊥ ∪ (a →3 b)) ∩ ((a →3 b)⊥ ∪ a)) = ((a →3 b)⊥ ∪ a)
31	21, 30	ax-r2 36	. . . 4 ((a →3 b)⊥ ∪ ((a →3 b) ∩ a)) = ((a →3 b)⊥ ∪ a)
32	20, 31	ax-r2 36	. . 3 ((((a →3 b)⊥ ∩ a) ∪ ((a →3 b)⊥ ∩ a⊥ )) ∪ ((a →3 b) ∩ ((a →3 b)⊥ ∪ a))) = ((a →3 b)⊥ ∪ a)
33	32	ax-r1 35	. 2 ((a →3 b)⊥ ∪ a) = ((((a →3 b)⊥ ∩ a) ∪ ((a →3 b)⊥ ∩ a⊥ )) ∪ ((a →3 b) ∩ ((a →3 b)⊥ ∪ a)))
34		lem4 511	. 2 ((a →3 b) →3 ((a →3 b) →3 a)) = ((a →3 b)⊥ ∪ a)
35		df-i3 46	. 2 ((a →3 b) →3 a) = ((((a →3 b)⊥ ∩ a) ∪ ((a →3 b)⊥ ∩ a⊥ )) ∪ ((a →3 b) ∩ ((a →3 b)⊥ ∪ a)))
36	33, 34, 35	3tr1 63	1 ((a →3 b) →3 ((a →3 b) →3 a)) = ((a →3 b) →3 a)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →₃ wi3 14

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  i3th7  549  i3th8  550