Theorem ud1lem2 543

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud1lem2	((a v (a_|_ ^ b_|_)) ->1 a) = (a v b)

Proof of Theorem ud1lem2

Step	Hyp	Ref	Expression
1		df-i1 43	. 2 ((a v (a_|_ ^ b_|_)) ->1 a) = ((a v (a_|_ ^ b_|_))_|_ v ((a v (a_|_ ^ b_|_)) ^ a))
2		comid 179	. . . . 5 (a v (a_|_ ^ b_|_)) C (a v (a_|_ ^ b_|_))
3	2	comcom3 436	. . . 4 (a v (a_|_ ^ b_|_))_|_ C (a v (a_|_ ^ b_|_))
4		comor1 443	. . . . 5 (a v (a_|_ ^ b_|_)) C a
5	4	comcom3 436	. . . 4 (a v (a_|_ ^ b_|_))_|_ C a
6	3, 5	fh3 453	. . 3 ((a v (a_|_ ^ b_|_))_|_ v ((a v (a_|_ ^ b_|_)) ^ a)) = (((a v (a_|_ ^ b_|_))_|_ v (a v (a_|_ ^ b_|_))) ^ ((a v (a_|_ ^ b_|_))_|_ v a))
7		ancom 68	. . . 4 (((a v (a_|_ ^ b_|_))_|_ v (a v (a_|_ ^ b_|_))) ^ ((a v (a_|_ ^ b_|_))_|_ v a)) = (((a v (a_|_ ^ b_|_))_|_ v a) ^ ((a v (a_|_ ^ b_|_))_|_ v (a v (a_|_ ^ b_|_))))
8		ax-a2 30	. . . . . . 7 ((a v (a_|_ ^ b_|_))_|_ v (a v (a_|_ ^ b_|_))) = ((a v (a_|_ ^ b_|_)) v (a v (a_|_ ^ b_|_))_|_)
9		df-t 40	. . . . . . . 8 1 = ((a v (a_|_ ^ b_|_)) v (a v (a_|_ ^ b_|_))_|_)
10	9	ax-r1 34	. . . . . . 7 ((a v (a_|_ ^ b_|_)) v (a v (a_|_ ^ b_|_))_|_) = 1
11	8, 10	ax-r2 35	. . . . . 6 ((a v (a_|_ ^ b_|_))_|_ v (a v (a_|_ ^ b_|_))) = 1
12	11	lan 70	. . . . 5 (((a v (a_|_ ^ b_|_))_|_ v a) ^ ((a v (a_|_ ^ b_|_))_|_ v (a v (a_|_ ^ b_|_)))) = (((a v (a_|_ ^ b_|_))_|_ v a) ^ 1)
13		an1 98	. . . . . 6 (((a v (a_|_ ^ b_|_))_|_ v a) ^ 1) = ((a v (a_|_ ^ b_|_))_|_ v a)
14		oran 79	. . . . . . . . . 10 (a v (a_|_ ^ b_|_)) = (a_|_ ^ (a_|_ ^ b_|_)_|_)_|_
15		oran 79	. . . . . . . . . . . . 13 (a v b) = (a_|_ ^ b_|_)_|_
16	15	ax-r1 34	. . . . . . . . . . . 12 (a_|_ ^ b_|_)_|_ = (a v b)
17	16	lan 70	. . . . . . . . . . 11 (a_|_ ^ (a_|_ ^ b_|_)_|_) = (a_|_ ^ (a v b))
18	17	ax-r4 36	. . . . . . . . . 10 (a_|_ ^ (a_|_ ^ b_|_)_|_)_|_ = (a_|_ ^ (a v b))_|_
19	14, 18	ax-r2 35	. . . . . . . . 9 (a v (a_|_ ^ b_|_)) = (a_|_ ^ (a v b))_|_
20	19	con2 64	. . . . . . . 8 (a v (a_|_ ^ b_|_))_|_ = (a_|_ ^ (a v b))
21	20	ax-r5 37	. . . . . . 7 ((a v (a_|_ ^ b_|_))_|_ v a) = ((a_|_ ^ (a v b)) v a)
22		ax-a2 30	. . . . . . . 8 ((a_|_ ^ (a v b)) v a) = (a v (a_|_ ^ (a v b)))
23		oml 427	. . . . . . . 8 (a v (a_|_ ^ (a v b))) = (a v b)
24	22, 23	ax-r2 35	. . . . . . 7 ((a_|_ ^ (a v b)) v a) = (a v b)
25	21, 24	ax-r2 35	. . . . . 6 ((a v (a_|_ ^ b_|_))_|_ v a) = (a v b)
26	13, 25	ax-r2 35	. . . . 5 (((a v (a_|_ ^ b_|_))_|_ v a) ^ 1) = (a v b)
27	12, 26	ax-r2 35	. . . 4 (((a v (a_|_ ^ b_|_))_|_ v a) ^ ((a v (a_|_ ^ b_|_))_|_ v (a v (a_|_ ^ b_|_)))) = (a v b)
28	7, 27	ax-r2 35	. . 3 (((a v (a_|_ ^ b_|_))_|_ v (a v (a_|_ ^ b_|_))) ^ ((a v (a_|_ ^ b_|_))_|_ v a)) = (a v b)
29	6, 28	ax-r2 35	. 2 ((a v (a_|_ ^ b_|_))_|_ v ((a v (a_|_ ^ b_|_)) ^ a)) = (a v b)
30	1, 29	ax-r2 35	1 ((a v (a_|_ ^ b_|_)) ->1 a) = (a v b)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->1 wi1 13

This theorem is referenced by:  ud1 577

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-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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