Theorem u3lem10 767

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u3lem10	(a ->3 (a_|_ ^ (a v b))) = a_|_

Proof of Theorem u3lem10

Step	Hyp	Ref	Expression
1		df-i3 45	. 2 (a ->3 (a_|_ ^ (a v b))) = (((a_|_ ^ (a_|_ ^ (a v b))) v (a_|_ ^ (a_|_ ^ (a v b))_|_)) v (a ^ (a_|_ v (a_|_ ^ (a v b)))))
2		anass 69	. . . . . . . 8 ((a_|_ ^ a_|_) ^ (a v b)) = (a_|_ ^ (a_|_ ^ (a v b)))
3	2	ax-r1 34	. . . . . . 7 (a_|_ ^ (a_|_ ^ (a v b))) = ((a_|_ ^ a_|_) ^ (a v b))
4		anidm 103	. . . . . . . 8 (a_|_ ^ a_|_) = a_|_
5	4	ran 71	. . . . . . 7 ((a_|_ ^ a_|_) ^ (a v b)) = (a_|_ ^ (a v b))
6	3, 5	ax-r2 35	. . . . . 6 (a_|_ ^ (a_|_ ^ (a v b))) = (a_|_ ^ (a v b))
7		anor3 82	. . . . . . . . . . 11 (a_|_ ^ b_|_) = (a v b)_|_
8	7	lor 66	. . . . . . . . . 10 (a v (a_|_ ^ b_|_)) = (a v (a v b)_|_)
9		oran1 83	. . . . . . . . . 10 (a v (a v b)_|_) = (a_|_ ^ (a v b))_|_
10	8, 9	ax-r2 35	. . . . . . . . 9 (a v (a_|_ ^ b_|_)) = (a_|_ ^ (a v b))_|_
11	10	ax-r1 34	. . . . . . . 8 (a_|_ ^ (a v b))_|_ = (a v (a_|_ ^ b_|_))
12	11	lan 70	. . . . . . 7 (a_|_ ^ (a_|_ ^ (a v b))_|_) = (a_|_ ^ (a v (a_|_ ^ b_|_)))
13		omlan 430	. . . . . . 7 (a_|_ ^ (a v (a_|_ ^ b_|_))) = (a_|_ ^ b_|_)
14	12, 13	ax-r2 35	. . . . . 6 (a_|_ ^ (a_|_ ^ (a v b))_|_) = (a_|_ ^ b_|_)
15	6, 14	2or 67	. . . . 5 ((a_|_ ^ (a_|_ ^ (a v b))) v (a_|_ ^ (a_|_ ^ (a v b))_|_)) = ((a_|_ ^ (a v b)) v (a_|_ ^ b_|_))
16		comanr1 446	. . . . . . 7 a_|_ C (a_|_ ^ b_|_)
17		comorr 176	. . . . . . . 8 a C (a v b)
18	17	comcom3 436	. . . . . . 7 a_|_ C (a v b)
19	16, 18	fh4r 458	. . . . . 6 ((a_|_ ^ (a v b)) v (a_|_ ^ b_|_)) = ((a_|_ v (a_|_ ^ b_|_)) ^ ((a v b) v (a_|_ ^ b_|_)))
20		a5b 112	. . . . . . . 8 (a_|_ v (a_|_ ^ b_|_)) = a_|_
21	7	lor 66	. . . . . . . . 9 ((a v b) v (a_|_ ^ b_|_)) = ((a v b) v (a v b)_|_)
22		df-t 40	. . . . . . . . . 10 1 = ((a v b) v (a v b)_|_)
23	22	ax-r1 34	. . . . . . . . 9 ((a v b) v (a v b)_|_) = 1
24	21, 23	ax-r2 35	. . . . . . . 8 ((a v b) v (a_|_ ^ b_|_)) = 1
25	20, 24	2an 72	. . . . . . 7 ((a_|_ v (a_|_ ^ b_|_)) ^ ((a v b) v (a_|_ ^ b_|_))) = (a_|_ ^ 1)
26		an1 98	. . . . . . 7 (a_|_ ^ 1) = a_|_
27	25, 26	ax-r2 35	. . . . . 6 ((a_|_ v (a_|_ ^ b_|_)) ^ ((a v b) v (a_|_ ^ b_|_))) = a_|_
28	19, 27	ax-r2 35	. . . . 5 ((a_|_ ^ (a v b)) v (a_|_ ^ b_|_)) = a_|_
29	15, 28	ax-r2 35	. . . 4 ((a_|_ ^ (a_|_ ^ (a v b))) v (a_|_ ^ (a_|_ ^ (a v b))_|_)) = a_|_
30		a5b 112	. . . . . 6 (a_|_ v (a_|_ ^ (a v b))) = a_|_
31	30	lan 70	. . . . 5 (a ^ (a_|_ v (a_|_ ^ (a v b)))) = (a ^ a_|_)
32		ancom 68	. . . . 5 (a ^ a_|_) = (a_|_ ^ a)
33	31, 32	ax-r2 35	. . . 4 (a ^ (a_|_ v (a_|_ ^ (a v b)))) = (a_|_ ^ a)
34	29, 33	2or 67	. . 3 (((a_|_ ^ (a_|_ ^ (a v b))) v (a_|_ ^ (a_|_ ^ (a v b))_|_)) v (a ^ (a_|_ v (a_|_ ^ (a v b))))) = (a_|_ v (a_|_ ^ a))
35		a5b 112	. . 3 (a_|_ v (a_|_ ^ a)) = a_|_
36	34, 35	ax-r2 35	. 2 (((a_|_ ^ (a_|_ ^ (a v b))) v (a_|_ ^ (a_|_ ^ (a v b))_|_)) v (a ^ (a_|_ v (a_|_ ^ (a v b))))) = a_|_
37	1, 36	ax-r2 35	1 (a ->3 (a_|_ ^ (a v b))) = a_|_

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->3 wi3 15

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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