Theorem u2lem8 764

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u2lem8	(a_|_ ->2 (a ->2 (a_|_ ->2 b))) = (a ->2 (a_|_ ->2 b))

Proof of Theorem u2lem8

Step	Hyp	Ref	Expression
1		df-i2 44	. 2 (a_|_ ->2 (a ->2 (a_|_ ->2 b))) = ((a ->2 (a_|_ ->2 b)) v (a_|__|_ ^ (a ->2 (a_|_ ->2 b))_|_))
2		u2lem7 755	. . . 4 (a ->2 (a_|_ ->2 b)) = (((a ^ b_|_) v (a_|_ ^ b_|_)) v b)
3		ax-a1 29	. . . . . . 7 a = a_|__|_
4	3	ax-r1 34	. . . . . 6 a_|__|_ = a
5		u2lem7n 757	. . . . . 6 (a ->2 (a_|_ ->2 b))_|_ = (((a v b) ^ (a_|_ v b)) ^ b_|_)
6	4, 5	2an 72	. . . . 5 (a_|__|_ ^ (a ->2 (a_|_ ->2 b))_|_) = (a ^ (((a v b) ^ (a_|_ v b)) ^ b_|_))
7		an12 74	. . . . . 6 (a ^ (((a v b) ^ (a_|_ v b)) ^ b_|_)) = (((a v b) ^ (a_|_ v b)) ^ (a ^ b_|_))
8		anass 69	. . . . . . 7 (((a v b) ^ (a_|_ v b)) ^ (a ^ b_|_)) = ((a v b) ^ ((a_|_ v b) ^ (a ^ b_|_)))
9		anor1 80	. . . . . . . . . . 11 (a ^ b_|_) = (a_|_ v b)_|_
10	9	lan 70	. . . . . . . . . 10 ((a_|_ v b) ^ (a ^ b_|_)) = ((a_|_ v b) ^ (a_|_ v b)_|_)
11		dff 93	. . . . . . . . . . 11 0 = ((a_|_ v b) ^ (a_|_ v b)_|_)
12	11	ax-r1 34	. . . . . . . . . 10 ((a_|_ v b) ^ (a_|_ v b)_|_) = 0
13	10, 12	ax-r2 35	. . . . . . . . 9 ((a_|_ v b) ^ (a ^ b_|_)) = 0
14	13	lan 70	. . . . . . . 8 ((a v b) ^ ((a_|_ v b) ^ (a ^ b_|_))) = ((a v b) ^ 0)
15		an0 100	. . . . . . . 8 ((a v b) ^ 0) = 0
16	14, 15	ax-r2 35	. . . . . . 7 ((a v b) ^ ((a_|_ v b) ^ (a ^ b_|_))) = 0
17	8, 16	ax-r2 35	. . . . . 6 (((a v b) ^ (a_|_ v b)) ^ (a ^ b_|_)) = 0
18	7, 17	ax-r2 35	. . . . 5 (a ^ (((a v b) ^ (a_|_ v b)) ^ b_|_)) = 0
19	6, 18	ax-r2 35	. . . 4 (a_|__|_ ^ (a ->2 (a_|_ ->2 b))_|_) = 0
20	2, 19	2or 67	. . 3 ((a ->2 (a_|_ ->2 b)) v (a_|__|_ ^ (a ->2 (a_|_ ->2 b))_|_)) = ((((a ^ b_|_) v (a_|_ ^ b_|_)) v b) v 0)
21		or0 94	. . . 4 ((((a ^ b_|_) v (a_|_ ^ b_|_)) v b) v 0) = (((a ^ b_|_) v (a_|_ ^ b_|_)) v b)
22	2	ax-r1 34	. . . 4 (((a ^ b_|_) v (a_|_ ^ b_|_)) v b) = (a ->2 (a_|_ ->2 b))
23	21, 22	ax-r2 35	. . 3 ((((a ^ b_|_) v (a_|_ ^ b_|_)) v b) v 0) = (a ->2 (a_|_ ->2 b))
24	20, 23	ax-r2 35	. 2 ((a ->2 (a_|_ ->2 b)) v (a_|__|_ ^ (a ->2 (a_|_ ->2 b))_|_)) = (a ->2 (a_|_ ->2 b))
25	1, 24	ax-r2 35	1 (a_|_ ->2 (a ->2 (a_|_ ->2 b))) = (a ->2 (a_|_ ->2 b))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  0wf 10   ->2 wi2 14

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

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i2 44  df-le1 122  df-le2 123

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