Theorem u4lem4 741

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u4lem4	(a ->4 (a ->4 (b ->4 a))) = (a ->4 (b ->4 a))

Proof of Theorem u4lem4

Step	Hyp	Ref	Expression
1		df-i4 46	. 2 (a ->4 (a ->4 (b ->4 a))) = (((a ^ (a ->4 (b ->4 a))) v (a_|_ ^ (a ->4 (b ->4 a)))) v ((a_|_ v (a ->4 (b ->4 a))) ^ (a ->4 (b ->4 a))_|_))
2		u4lem3 734	. . . . . . . . 9 (a ->4 (b ->4 a)) = (a_|_ v ((a ^ b) v (a ^ b_|_)))
3		comid 179	. . . . . . . . . . . 12 a C a
4	3	comcom2 175	. . . . . . . . . . 11 a C a_|_
5		comanr1 446	. . . . . . . . . . . 12 a C (a ^ b)
6		comanr1 446	. . . . . . . . . . . 12 a C (a ^ b_|_)
7	5, 6	com2or 465	. . . . . . . . . . 11 a C ((a ^ b) v (a ^ b_|_))
8	4, 7	com2or 465	. . . . . . . . . 10 a C (a_|_ v ((a ^ b) v (a ^ b_|_)))
9	8	comcom 435	. . . . . . . . 9 (a_|_ v ((a ^ b) v (a ^ b_|_))) C a
10	2, 9	bctr 173	. . . . . . . 8 (a ->4 (b ->4 a)) C a
11	10	comcom 435	. . . . . . 7 a C (a ->4 (b ->4 a))
12	11, 4	fh2r 456	. . . . . 6 ((a v a_|_) ^ (a ->4 (b ->4 a))) = ((a ^ (a ->4 (b ->4 a))) v (a_|_ ^ (a ->4 (b ->4 a))))
13	12	ax-r1 34	. . . . 5 ((a ^ (a ->4 (b ->4 a))) v (a_|_ ^ (a ->4 (b ->4 a)))) = ((a v a_|_) ^ (a ->4 (b ->4 a)))
14		ancom 68	. . . . . 6 ((a v a_|_) ^ (a ->4 (b ->4 a))) = ((a ->4 (b ->4 a)) ^ (a v a_|_))
15		df-t 40	. . . . . . . . 9 1 = (a v a_|_)
16	15	ax-r1 34	. . . . . . . 8 (a v a_|_) = 1
17	16	lan 70	. . . . . . 7 ((a ->4 (b ->4 a)) ^ (a v a_|_)) = ((a ->4 (b ->4 a)) ^ 1)
18		an1 98	. . . . . . 7 ((a ->4 (b ->4 a)) ^ 1) = (a ->4 (b ->4 a))
19	17, 18	ax-r2 35	. . . . . 6 ((a ->4 (b ->4 a)) ^ (a v a_|_)) = (a ->4 (b ->4 a))
20	14, 19	ax-r2 35	. . . . 5 ((a v a_|_) ^ (a ->4 (b ->4 a))) = (a ->4 (b ->4 a))
21	13, 20	ax-r2 35	. . . 4 ((a ^ (a ->4 (b ->4 a))) v (a_|_ ^ (a ->4 (b ->4 a)))) = (a ->4 (b ->4 a))
22	10	comcom4 437	. . . . . 6 (a ->4 (b ->4 a))_|_ C a_|_
23		comid 179	. . . . . . 7 (a ->4 (b ->4 a)) C (a ->4 (b ->4 a))
24	23	comcom3 436	. . . . . 6 (a ->4 (b ->4 a))_|_ C (a ->4 (b ->4 a))
25	22, 24	fh1r 455	. . . . 5 ((a_|_ v (a ->4 (b ->4 a))) ^ (a ->4 (b ->4 a))_|_) = ((a_|_ ^ (a ->4 (b ->4 a))_|_) v ((a ->4 (b ->4 a)) ^ (a ->4 (b ->4 a))_|_))
26		dff 93	. . . . . . . 8 0 = ((a ->4 (b ->4 a)) ^ (a ->4 (b ->4 a))_|_)
27	26	ax-r1 34	. . . . . . 7 ((a ->4 (b ->4 a)) ^ (a ->4 (b ->4 a))_|_) = 0
28	27	lor 66	. . . . . 6 ((a_|_ ^ (a ->4 (b ->4 a))_|_) v ((a ->4 (b ->4 a)) ^ (a ->4 (b ->4 a))_|_)) = ((a_|_ ^ (a ->4 (b ->4 a))_|_) v 0)
29		or0 94	. . . . . 6 ((a_|_ ^ (a ->4 (b ->4 a))_|_) v 0) = (a_|_ ^ (a ->4 (b ->4 a))_|_)
30	28, 29	ax-r2 35	. . . . 5 ((a_|_ ^ (a ->4 (b ->4 a))_|_) v ((a ->4 (b ->4 a)) ^ (a ->4 (b ->4 a))_|_)) = (a_|_ ^ (a ->4 (b ->4 a))_|_)
31	25, 30	ax-r2 35	. . . 4 ((a_|_ v (a ->4 (b ->4 a))) ^ (a ->4 (b ->4 a))_|_) = (a_|_ ^ (a ->4 (b ->4 a))_|_)
32	21, 31	2or 67	. . 3 (((a ^ (a ->4 (b ->4 a))) v (a_|_ ^ (a ->4 (b ->4 a)))) v ((a_|_ v (a ->4 (b ->4 a))) ^ (a ->4 (b ->4 a))_|_)) = ((a ->4 (b ->4 a)) v (a_|_ ^ (a ->4 (b ->4 a))_|_))
33	10	comcom2 175	. . . . . 6 (a ->4 (b ->4 a)) C a_|_
34	23	comcom2 175	. . . . . 6 (a ->4 (b ->4 a)) C (a ->4 (b ->4 a))_|_
35	33, 34	fh3 453	. . . . 5 ((a ->4 (b ->4 a)) v (a_|_ ^ (a ->4 (b ->4 a))_|_)) = (((a ->4 (b ->4 a)) v a_|_) ^ ((a ->4 (b ->4 a)) v (a ->4 (b ->4 a))_|_))
36		df-t 40	. . . . . . . 8 1 = ((a ->4 (b ->4 a)) v (a ->4 (b ->4 a))_|_)
37	36	ax-r1 34	. . . . . . 7 ((a ->4 (b ->4 a)) v (a ->4 (b ->4 a))_|_) = 1
38	37	lan 70	. . . . . 6 (((a ->4 (b ->4 a)) v a_|_) ^ ((a ->4 (b ->4 a)) v (a ->4 (b ->4 a))_|_)) = (((a ->4 (b ->4 a)) v a_|_) ^ 1)
39		an1 98	. . . . . 6 (((a ->4 (b ->4 a)) v a_|_) ^ 1) = ((a ->4 (b ->4 a)) v a_|_)
40	38, 39	ax-r2 35	. . . . 5 (((a ->4 (b ->4 a)) v a_|_) ^ ((a ->4 (b ->4 a)) v (a ->4 (b ->4 a))_|_)) = ((a ->4 (b ->4 a)) v a_|_)
41	35, 40	ax-r2 35	. . . 4 ((a ->4 (b ->4 a)) v (a_|_ ^ (a ->4 (b ->4 a))_|_)) = ((a ->4 (b ->4 a)) v a_|_)
42	2	ax-r5 37	. . . . 5 ((a ->4 (b ->4 a)) v a_|_) = ((a_|_ v ((a ^ b) v (a ^ b_|_))) v a_|_)
43		or32 75	. . . . . 6 ((a_|_ v ((a ^ b) v (a ^ b_|_))) v a_|_) = ((a_|_ v a_|_) v ((a ^ b) v (a ^ b_|_)))
44		oridm 102	. . . . . . . 8 (a_|_ v a_|_) = a_|_
45	44	ax-r5 37	. . . . . . 7 ((a_|_ v a_|_) v ((a ^ b) v (a ^ b_|_))) = (a_|_ v ((a ^ b) v (a ^ b_|_)))
46	2	ax-r1 34	. . . . . . 7 (a_|_ v ((a ^ b) v (a ^ b_|_))) = (a ->4 (b ->4 a))
47	45, 46	ax-r2 35	. . . . . 6 ((a_|_ v a_|_) v ((a ^ b) v (a ^ b_|_))) = (a ->4 (b ->4 a))
48	43, 47	ax-r2 35	. . . . 5 ((a_|_ v ((a ^ b) v (a ^ b_|_))) v a_|_) = (a ->4 (b ->4 a))
49	42, 48	ax-r2 35	. . . 4 ((a ->4 (b ->4 a)) v a_|_) = (a ->4 (b ->4 a))
50	41, 49	ax-r2 35	. . 3 ((a ->4 (b ->4 a)) v (a_|_ ^ (a ->4 (b ->4 a))_|_)) = (a ->4 (b ->4 a))
51	32, 50	ax-r2 35	. 2 (((a ^ (a ->4 (b ->4 a))) v (a_|_ ^ (a ->4 (b ->4 a)))) v ((a_|_ v (a ->4 (b ->4 a))) ^ (a ->4 (b ->4 a))_|_)) = (a ->4 (b ->4 a))
52	1, 51	ax-r2 35	1 (a ->4 (a ->4 (b ->4 a))) = (a ->4 (b ->4 a))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9  0wf 10   ->4 wi4 16

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-i4 46  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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