Theorem u5lem6 751

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u5lem6	(a ->5 (a ->5 (a ->5 b))) = (a ->5 (a ->5 b))

Proof of Theorem u5lem6

Step	Hyp	Ref	Expression
1		df-i5 47	. 2 (a ->5 (a ->5 (a ->5 b))) = (((a ^ (a ->5 (a ->5 b))) v (a_|_ ^ (a ->5 (a ->5 b)))) v (a_|_ ^ (a ->5 (a ->5 b))_|_))
2		ancom 68	. . . . 5 ((a v a_|_) ^ (a ->5 (a ->5 b))) = ((a ->5 (a ->5 b)) ^ (a v a_|_))
3		u5lemc1 666	. . . . . . 7 a C (a ->5 (a ->5 b))
4	3	comcom 435	. . . . . 6 (a ->5 (a ->5 b)) C a
5	4	comcom2 175	. . . . . 6 (a ->5 (a ->5 b)) C a_|_
6	4, 5	fh1r 455	. . . . 5 ((a v a_|_) ^ (a ->5 (a ->5 b))) = ((a ^ (a ->5 (a ->5 b))) v (a_|_ ^ (a ->5 (a ->5 b))))
7		df-t 40	. . . . . . . 8 1 = (a v a_|_)
8	7	ax-r1 34	. . . . . . 7 (a v a_|_) = 1
9	8	lan 70	. . . . . 6 ((a ->5 (a ->5 b)) ^ (a v a_|_)) = ((a ->5 (a ->5 b)) ^ 1)
10		an1 98	. . . . . 6 ((a ->5 (a ->5 b)) ^ 1) = (a ->5 (a ->5 b))
11	9, 10	ax-r2 35	. . . . 5 ((a ->5 (a ->5 b)) ^ (a v a_|_)) = (a ->5 (a ->5 b))
12	2, 6, 11	3tr2 61	. . . 4 ((a ^ (a ->5 (a ->5 b))) v (a_|_ ^ (a ->5 (a ->5 b)))) = (a ->5 (a ->5 b))
13	12	ax-r5 37	. . 3 (((a ^ (a ->5 (a ->5 b))) v (a_|_ ^ (a ->5 (a ->5 b)))) v (a_|_ ^ (a ->5 (a ->5 b))_|_)) = ((a ->5 (a ->5 b)) v (a_|_ ^ (a ->5 (a ->5 b))_|_))
14	3	comcom3 436	. . . . 5 a_|_ C (a ->5 (a ->5 b))
15	3	comcom4 437	. . . . 5 a_|_ C (a ->5 (a ->5 b))_|_
16	14, 15	fh4 454	. . . 4 ((a ->5 (a ->5 b)) v (a_|_ ^ (a ->5 (a ->5 b))_|_)) = (((a ->5 (a ->5 b)) v a_|_) ^ ((a ->5 (a ->5 b)) v (a ->5 (a ->5 b))_|_))
17		df-t 40	. . . . . . 7 1 = ((a ->5 (a ->5 b)) v (a ->5 (a ->5 b))_|_)
18	17	ax-r1 34	. . . . . 6 ((a ->5 (a ->5 b)) v (a ->5 (a ->5 b))_|_) = 1
19	18	lan 70	. . . . 5 (((a ->5 (a ->5 b)) v a_|_) ^ ((a ->5 (a ->5 b)) v (a ->5 (a ->5 b))_|_)) = (((a ->5 (a ->5 b)) v a_|_) ^ 1)
20		an1 98	. . . . . 6 (((a ->5 (a ->5 b)) v a_|_) ^ 1) = ((a ->5 (a ->5 b)) v a_|_)
21		u5lem5 747	. . . . . . . 8 (a ->5 (a ->5 b)) = (a_|_ v (a ^ b))
22	21	ax-r5 37	. . . . . . 7 ((a ->5 (a ->5 b)) v a_|_) = ((a_|_ v (a ^ b)) v a_|_)
23		oridm 102	. . . . . . . . 9 (a_|_ v a_|_) = a_|_
24	23	ax-r5 37	. . . . . . . 8 ((a_|_ v a_|_) v (a ^ b)) = (a_|_ v (a ^ b))
25		or32 75	. . . . . . . 8 ((a_|_ v (a ^ b)) v a_|_) = ((a_|_ v a_|_) v (a ^ b))
26	24, 25, 21	3tr1 60	. . . . . . 7 ((a_|_ v (a ^ b)) v a_|_) = (a ->5 (a ->5 b))
27	22, 26	ax-r2 35	. . . . . 6 ((a ->5 (a ->5 b)) v a_|_) = (a ->5 (a ->5 b))
28	20, 27	ax-r2 35	. . . . 5 (((a ->5 (a ->5 b)) v a_|_) ^ 1) = (a ->5 (a ->5 b))
29	19, 28	ax-r2 35	. . . 4 (((a ->5 (a ->5 b)) v a_|_) ^ ((a ->5 (a ->5 b)) v (a ->5 (a ->5 b))_|_)) = (a ->5 (a ->5 b))
30	16, 29	ax-r2 35	. . 3 ((a ->5 (a ->5 b)) v (a_|_ ^ (a ->5 (a ->5 b))_|_)) = (a ->5 (a ->5 b))
31	13, 30	ax-r2 35	. 2 (((a ^ (a ->5 (a ->5 b))) v (a_|_ ^ (a ->5 (a ->5 b)))) v (a_|_ ^ (a ->5 (a ->5 b))_|_)) = (a ->5 (a ->5 b))
32	1, 31	ax-r2 35	1 (a ->5 (a ->5 (a ->5 b))) = (a ->5 (a ->5 b))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->5 wi5 17

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-i5 47  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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