Theorem ud2lem2 546

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud2lem2	((a v (a_|_ ^ b_|_)) ->2 a) = (a v b)

Proof of Theorem ud2lem2

Step	Hyp	Ref	Expression
1		df-i2 44	. 2 ((a v (a_|_ ^ b_|_)) ->2 a) = (a v ((a v (a_|_ ^ b_|_))_|_ ^ a_|_))
2		oran 79	. . . . . . 7 ((a v (a_|_ ^ b_|_)) v a) = ((a v (a_|_ ^ b_|_))_|_ ^ a_|_)_|_
3	2	con2 64	. . . . . 6 ((a v (a_|_ ^ b_|_)) v a)_|_ = ((a v (a_|_ ^ b_|_))_|_ ^ a_|_)
4	3	ax-r1 34	. . . . 5 ((a v (a_|_ ^ b_|_))_|_ ^ a_|_) = ((a v (a_|_ ^ b_|_)) v a)_|_
5		oran 79	. . . . . . . . . . . . 13 (a v b) = (a_|_ ^ b_|_)_|_
6	5	con2 64	. . . . . . . . . . . 12 (a v b)_|_ = (a_|_ ^ b_|_)
7	6	ax-r1 34	. . . . . . . . . . 11 (a_|_ ^ b_|_) = (a v b)_|_
8	7	lor 66	. . . . . . . . . 10 (a v (a_|_ ^ b_|_)) = (a v (a v b)_|_)
9		anor2 81	. . . . . . . . . . . 12 (a_|_ ^ (a v b)) = (a v (a v b)_|_)_|_
10	9	ax-r1 34	. . . . . . . . . . 11 (a v (a v b)_|_)_|_ = (a_|_ ^ (a v b))
11	10	con3 65	. . . . . . . . . 10 (a v (a v b)_|_) = (a_|_ ^ (a v b))_|_
12	8, 11	ax-r2 35	. . . . . . . . 9 (a v (a_|_ ^ b_|_)) = (a_|_ ^ (a v b))_|_
13	12	con2 64	. . . . . . . 8 (a v (a_|_ ^ b_|_))_|_ = (a_|_ ^ (a v b))
14	13	ran 71	. . . . . . 7 ((a v (a_|_ ^ b_|_))_|_ ^ a_|_) = ((a_|_ ^ (a v b)) ^ a_|_)
15		an32 76	. . . . . . . 8 ((a_|_ ^ (a v b)) ^ a_|_) = ((a_|_ ^ a_|_) ^ (a v b))
16		anidm 103	. . . . . . . . 9 (a_|_ ^ a_|_) = a_|_
17	16	ran 71	. . . . . . . 8 ((a_|_ ^ a_|_) ^ (a v b)) = (a_|_ ^ (a v b))
18	15, 17	ax-r2 35	. . . . . . 7 ((a_|_ ^ (a v b)) ^ a_|_) = (a_|_ ^ (a v b))
19	14, 18	ax-r2 35	. . . . . 6 ((a v (a_|_ ^ b_|_))_|_ ^ a_|_) = (a_|_ ^ (a v b))
20	3, 19	ax-r2 35	. . . . 5 ((a v (a_|_ ^ b_|_)) v a)_|_ = (a_|_ ^ (a v b))
21	4, 20	ax-r2 35	. . . 4 ((a v (a_|_ ^ b_|_))_|_ ^ a_|_) = (a_|_ ^ (a v b))
22	21	lor 66	. . 3 (a v ((a v (a_|_ ^ b_|_))_|_ ^ a_|_)) = (a v (a_|_ ^ (a v b)))
23		oml 427	. . 3 (a v (a_|_ ^ (a v b))) = (a v b)
24	22, 23	ax-r2 35	. 2 (a v ((a v (a_|_ ^ b_|_))_|_ ^ a_|_)) = (a v b)
25	1, 24	ax-r2 35	1 ((a v (a_|_ ^ b_|_)) ->2 a) = (a v b)

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

This theorem is referenced by:  ud2 578

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  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-i2 44

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