Theorem i3abs3 506

Description: Antecedent absorption.

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

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9  0wf 10   ->3 wi3 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

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