Theorem ud4lem2 564

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud4lem2	((a v (a_|_ ^ b_|_)) ->4 a) = (a v b)

Proof of Theorem ud4lem2

Step	Hyp	Ref	Expression
1		df-i4 46	. 2 ((a v (a_|_ ^ b_|_)) ->4 a) = ((((a v (a_|_ ^ b_|_)) ^ a) v ((a v (a_|_ ^ b_|_))_|_ ^ a)) v (((a v (a_|_ ^ b_|_))_|_ v a) ^ a_|_))
2		ancom 68	. . . . . . 7 ((a v (a_|_ ^ b_|_)) ^ a) = (a ^ (a v (a_|_ ^ b_|_)))
3		a5c 113	. . . . . . 7 (a ^ (a v (a_|_ ^ b_|_))) = a
4	2, 3	ax-r2 35	. . . . . 6 ((a v (a_|_ ^ b_|_)) ^ a) = a
5		oran 79	. . . . . . . . 9 (a v (a_|_ ^ b_|_)) = (a_|_ ^ (a_|_ ^ b_|_)_|_)_|_
6	5	con2 64	. . . . . . . 8 (a v (a_|_ ^ b_|_))_|_ = (a_|_ ^ (a_|_ ^ b_|_)_|_)
7	6	ran 71	. . . . . . 7 ((a v (a_|_ ^ b_|_))_|_ ^ a) = ((a_|_ ^ (a_|_ ^ b_|_)_|_) ^ a)
8		ancom 68	. . . . . . . 8 ((a_|_ ^ (a_|_ ^ b_|_)_|_) ^ a) = (a ^ (a_|_ ^ (a_|_ ^ b_|_)_|_))
9		anass 69	. . . . . . . . . 10 ((a ^ a_|_) ^ (a_|_ ^ b_|_)_|_) = (a ^ (a_|_ ^ (a_|_ ^ b_|_)_|_))
10	9	ax-r1 34	. . . . . . . . 9 (a ^ (a_|_ ^ (a_|_ ^ b_|_)_|_)) = ((a ^ a_|_) ^ (a_|_ ^ b_|_)_|_)
11		ancom 68	. . . . . . . . . 10 ((a ^ a_|_) ^ (a_|_ ^ b_|_)_|_) = ((a_|_ ^ b_|_)_|_ ^ (a ^ a_|_))
12		dff 93	. . . . . . . . . . . . 13 0 = (a ^ a_|_)
13	12	lan 70	. . . . . . . . . . . 12 ((a_|_ ^ b_|_)_|_ ^ 0) = ((a_|_ ^ b_|_)_|_ ^ (a ^ a_|_))
14	13	ax-r1 34	. . . . . . . . . . 11 ((a_|_ ^ b_|_)_|_ ^ (a ^ a_|_)) = ((a_|_ ^ b_|_)_|_ ^ 0)
15		an0 100	. . . . . . . . . . 11 ((a_|_ ^ b_|_)_|_ ^ 0) = 0
16	14, 15	ax-r2 35	. . . . . . . . . 10 ((a_|_ ^ b_|_)_|_ ^ (a ^ a_|_)) = 0
17	11, 16	ax-r2 35	. . . . . . . . 9 ((a ^ a_|_) ^ (a_|_ ^ b_|_)_|_) = 0
18	10, 17	ax-r2 35	. . . . . . . 8 (a ^ (a_|_ ^ (a_|_ ^ b_|_)_|_)) = 0
19	8, 18	ax-r2 35	. . . . . . 7 ((a_|_ ^ (a_|_ ^ b_|_)_|_) ^ a) = 0
20	7, 19	ax-r2 35	. . . . . 6 ((a v (a_|_ ^ b_|_))_|_ ^ a) = 0
21	4, 20	2or 67	. . . . 5 (((a v (a_|_ ^ b_|_)) ^ a) v ((a v (a_|_ ^ b_|_))_|_ ^ a)) = (a v 0)
22		or0 94	. . . . 5 (a v 0) = a
23	21, 22	ax-r2 35	. . . 4 (((a v (a_|_ ^ b_|_)) ^ a) v ((a v (a_|_ ^ b_|_))_|_ ^ a)) = a
24		ancom 68	. . . . 5 (((a v (a_|_ ^ b_|_))_|_ v a) ^ a_|_) = (a_|_ ^ ((a v (a_|_ ^ b_|_))_|_ v a))
25		oran 79	. . . . . . . . . . . . 13 (a v b) = (a_|_ ^ b_|_)_|_
26	25	ax-r1 34	. . . . . . . . . . . 12 (a_|_ ^ b_|_)_|_ = (a v b)
27	26	con3 65	. . . . . . . . . . 11 (a_|_ ^ b_|_) = (a v b)_|_
28	27	lor 66	. . . . . . . . . 10 (a v (a_|_ ^ b_|_)) = (a v (a v b)_|_)
29		anor2 81	. . . . . . . . . . . 12 (a_|_ ^ (a v b)) = (a v (a v b)_|_)_|_
30	29	ax-r1 34	. . . . . . . . . . 11 (a v (a v b)_|_)_|_ = (a_|_ ^ (a v b))
31	30	con3 65	. . . . . . . . . 10 (a v (a v b)_|_) = (a_|_ ^ (a v b))_|_
32	28, 31	ax-r2 35	. . . . . . . . 9 (a v (a_|_ ^ b_|_)) = (a_|_ ^ (a v b))_|_
33	32	con2 64	. . . . . . . 8 (a v (a_|_ ^ b_|_))_|_ = (a_|_ ^ (a v b))
34	33	ax-r5 37	. . . . . . 7 ((a v (a_|_ ^ b_|_))_|_ v a) = ((a_|_ ^ (a v b)) v a)
35		comid 179	. . . . . . . . . 10 a C a
36	35	comcom2 175	. . . . . . . . 9 a C a_|_
37		comorr 176	. . . . . . . . 9 a C (a v b)
38	36, 37	fh3r 457	. . . . . . . 8 ((a_|_ ^ (a v b)) v a) = ((a_|_ v a) ^ ((a v b) v a))
39		ancom 68	. . . . . . . . . 10 ((a_|_ v a) ^ ((a v b) v a)) = (((a v b) v a) ^ (a_|_ v a))
40		or32 75	. . . . . . . . . . . 12 ((a v b) v a) = ((a v a) v b)
41		oridm 102	. . . . . . . . . . . . 13 (a v a) = a
42	41	ax-r5 37	. . . . . . . . . . . 12 ((a v a) v b) = (a v b)
43	40, 42	ax-r2 35	. . . . . . . . . . 11 ((a v b) v a) = (a v b)
44		df-t 40	. . . . . . . . . . . . 13 1 = (a v a_|_)
45		ax-a2 30	. . . . . . . . . . . . 13 (a v a_|_) = (a_|_ v a)
46	44, 45	ax-r2 35	. . . . . . . . . . . 12 1 = (a_|_ v a)
47	46	ax-r1 34	. . . . . . . . . . 11 (a_|_ v a) = 1
48	43, 47	2an 72	. . . . . . . . . 10 (((a v b) v a) ^ (a_|_ v a)) = ((a v b) ^ 1)
49	39, 48	ax-r2 35	. . . . . . . . 9 ((a_|_ v a) ^ ((a v b) v a)) = ((a v b) ^ 1)
50		an1 98	. . . . . . . . 9 ((a v b) ^ 1) = (a v b)
51	49, 50	ax-r2 35	. . . . . . . 8 ((a_|_ v a) ^ ((a v b) v a)) = (a v b)
52	38, 51	ax-r2 35	. . . . . . 7 ((a_|_ ^ (a v b)) v a) = (a v b)
53	34, 52	ax-r2 35	. . . . . 6 ((a v (a_|_ ^ b_|_))_|_ v a) = (a v b)
54	53	lan 70	. . . . 5 (a_|_ ^ ((a v (a_|_ ^ b_|_))_|_ v a)) = (a_|_ ^ (a v b))
55	24, 54	ax-r2 35	. . . 4 (((a v (a_|_ ^ b_|_))_|_ v a) ^ a_|_) = (a_|_ ^ (a v b))
56	23, 55	2or 67	. . 3 ((((a v (a_|_ ^ b_|_)) ^ a) v ((a v (a_|_ ^ b_|_))_|_ ^ a)) v (((a v (a_|_ ^ b_|_))_|_ v a) ^ a_|_)) = (a v (a_|_ ^ (a v b)))
57		oml 427	. . 3 (a v (a_|_ ^ (a v b))) = (a v b)
58	56, 57	ax-r2 35	. 2 ((((a v (a_|_ ^ b_|_)) ^ a) v ((a v (a_|_ ^ b_|_))_|_ ^ a)) v (((a v (a_|_ ^ b_|_))_|_ v a) ^ a_|_)) = (a v b)
59	1, 58	ax-r2 35	1 ((a v (a_|_ ^ b_|_)) ->4 a) = (a v b)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9  0wf 10   ->4 wi4 16

This theorem is referenced by:  ud4 580

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-i4 46  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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