Theorem ud5lem3b 574

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud5lem3b	((a ->5 b)_|_ ^ (a v (a_|_ ^ b))) = (a ^ (a_|_ v b_|_))

Proof of Theorem ud5lem3b

Step	Hyp	Ref	Expression
1		ud5lem0c 273	. . 3 (a ->5 b)_|_ = (((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b))
2	1	ran 71	. 2 ((a ->5 b)_|_ ^ (a v (a_|_ ^ b))) = ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b)) ^ (a v (a_|_ ^ b)))
3		comorr 176	. . . . . . 7 a_|_ C (a_|_ v b_|_)
4	3	comcom6 441	. . . . . 6 a C (a_|_ v b_|_)
5		comorr 176	. . . . . 6 a C (a v b_|_)
6	4, 5	com2an 466	. . . . 5 a C ((a_|_ v b_|_) ^ (a v b_|_))
7		comorr 176	. . . . 5 a C (a v b)
8	6, 7	com2an 466	. . . 4 a C (((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b))
9		comanr1 446	. . . . 5 a_|_ C (a_|_ ^ b)
10	9	comcom6 441	. . . 4 a C (a_|_ ^ b)
11	8, 10	fh2 452	. . 3 ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b)) ^ (a v (a_|_ ^ b))) = (((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b)) ^ a) v ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b)) ^ (a_|_ ^ b)))
12		anass 69	. . . . . 6 ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b)) ^ a) = (((a_|_ v b_|_) ^ (a v b_|_)) ^ ((a v b) ^ a))
13		ancom 68	. . . . . . . . 9 ((a v b) ^ a) = (a ^ (a v b))
14		a5c 113	. . . . . . . . 9 (a ^ (a v b)) = a
15	13, 14	ax-r2 35	. . . . . . . 8 ((a v b) ^ a) = a
16	15	lan 70	. . . . . . 7 (((a_|_ v b_|_) ^ (a v b_|_)) ^ ((a v b) ^ a)) = (((a_|_ v b_|_) ^ (a v b_|_)) ^ a)
17		anass 69	. . . . . . . 8 (((a_|_ v b_|_) ^ (a v b_|_)) ^ a) = ((a_|_ v b_|_) ^ ((a v b_|_) ^ a))
18		ancom 68	. . . . . . . . . . 11 ((a v b_|_) ^ a) = (a ^ (a v b_|_))
19		a5c 113	. . . . . . . . . . 11 (a ^ (a v b_|_)) = a
20	18, 19	ax-r2 35	. . . . . . . . . 10 ((a v b_|_) ^ a) = a
21	20	lan 70	. . . . . . . . 9 ((a_|_ v b_|_) ^ ((a v b_|_) ^ a)) = ((a_|_ v b_|_) ^ a)
22		ancom 68	. . . . . . . . 9 ((a_|_ v b_|_) ^ a) = (a ^ (a_|_ v b_|_))
23	21, 22	ax-r2 35	. . . . . . . 8 ((a_|_ v b_|_) ^ ((a v b_|_) ^ a)) = (a ^ (a_|_ v b_|_))
24	17, 23	ax-r2 35	. . . . . . 7 (((a_|_ v b_|_) ^ (a v b_|_)) ^ a) = (a ^ (a_|_ v b_|_))
25	16, 24	ax-r2 35	. . . . . 6 (((a_|_ v b_|_) ^ (a v b_|_)) ^ ((a v b) ^ a)) = (a ^ (a_|_ v b_|_))
26	12, 25	ax-r2 35	. . . . 5 ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b)) ^ a) = (a ^ (a_|_ v b_|_))
27		an32 76	. . . . . 6 ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b)) ^ (a_|_ ^ b)) = ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a_|_ ^ b)) ^ (a v b))
28		anass 69	. . . . . . . . 9 (((a_|_ v b_|_) ^ (a v b_|_)) ^ (a_|_ ^ b)) = ((a_|_ v b_|_) ^ ((a v b_|_) ^ (a_|_ ^ b)))
29		anor2 81	. . . . . . . . . . . . 13 (a_|_ ^ b) = (a v b_|_)_|_
30	29	lan 70	. . . . . . . . . . . 12 ((a v b_|_) ^ (a_|_ ^ b)) = ((a v b_|_) ^ (a v b_|_)_|_)
31		dff 93	. . . . . . . . . . . . 13 0 = ((a v b_|_) ^ (a v b_|_)_|_)
32	31	ax-r1 34	. . . . . . . . . . . 12 ((a v b_|_) ^ (a v b_|_)_|_) = 0
33	30, 32	ax-r2 35	. . . . . . . . . . 11 ((a v b_|_) ^ (a_|_ ^ b)) = 0
34	33	lan 70	. . . . . . . . . 10 ((a_|_ v b_|_) ^ ((a v b_|_) ^ (a_|_ ^ b))) = ((a_|_ v b_|_) ^ 0)
35		an0 100	. . . . . . . . . 10 ((a_|_ v b_|_) ^ 0) = 0
36	34, 35	ax-r2 35	. . . . . . . . 9 ((a_|_ v b_|_) ^ ((a v b_|_) ^ (a_|_ ^ b))) = 0
37	28, 36	ax-r2 35	. . . . . . . 8 (((a_|_ v b_|_) ^ (a v b_|_)) ^ (a_|_ ^ b)) = 0
38	37	ran 71	. . . . . . 7 ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a_|_ ^ b)) ^ (a v b)) = (0 ^ (a v b))
39		an0r 101	. . . . . . 7 (0 ^ (a v b)) = 0
40	38, 39	ax-r2 35	. . . . . 6 ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a_|_ ^ b)) ^ (a v b)) = 0
41	27, 40	ax-r2 35	. . . . 5 ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b)) ^ (a_|_ ^ b)) = 0
42	26, 41	2or 67	. . . 4 (((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b)) ^ a) v ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b)) ^ (a_|_ ^ b))) = ((a ^ (a_|_ v b_|_)) v 0)
43		or0 94	. . . 4 ((a ^ (a_|_ v b_|_)) v 0) = (a ^ (a_|_ v b_|_))
44	42, 43	ax-r2 35	. . 3 (((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b)) ^ a) v ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b)) ^ (a_|_ ^ b))) = (a ^ (a_|_ v b_|_))
45	11, 44	ax-r2 35	. 2 ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b)) ^ (a v (a_|_ ^ b))) = (a ^ (a_|_ v b_|_))
46	2, 45	ax-r2 35	1 ((a ->5 b)_|_ ^ (a v (a_|_ ^ b))) = (a ^ (a_|_ v b_|_))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  0wf 10   ->5 wi5 17

This theorem is referenced by:  ud5lem3 576

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-i5 47  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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