Theorem ud3lem1c 550

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud3lem1c	((a ->3 b)_|_ v (b ->3 a)) = (a v b_|_)

Proof of Theorem ud3lem1c

Step	Hyp	Ref	Expression
1		ud3lem0c 271	. . 3 (a ->3 b)_|_ = (((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_)))
2		df-i3 45	. . 3 (b ->3 a) = (((b_|_ ^ a) v (b_|_ ^ a_|_)) v (b ^ (b_|_ v a)))
3	1, 2	2or 67	. 2 ((a ->3 b)_|_ v (b ->3 a)) = ((((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_))) v (((b_|_ ^ a) v (b_|_ ^ a_|_)) v (b ^ (b_|_ v a))))
4		coman2 178	. . . . . . . . 9 (b_|_ ^ a) C a
5		coman1 177	. . . . . . . . 9 (b_|_ ^ a) C b_|_
6	4, 5	com2or 465	. . . . . . . 8 (b_|_ ^ a) C (a v b_|_)
7	5	comcom7 442	. . . . . . . . 9 (b_|_ ^ a) C b
8	4, 7	com2or 465	. . . . . . . 8 (b_|_ ^ a) C (a v b)
9	6, 8	com2an 466	. . . . . . 7 (b_|_ ^ a) C ((a v b_|_) ^ (a v b))
10	9	comcom 435	. . . . . 6 ((a v b_|_) ^ (a v b)) C (b_|_ ^ a)
11		coman2 178	. . . . . . . . . 10 (b_|_ ^ a_|_) C a_|_
12	11	comcom7 442	. . . . . . . . 9 (b_|_ ^ a_|_) C a
13		coman1 177	. . . . . . . . 9 (b_|_ ^ a_|_) C b_|_
14	12, 13	com2or 465	. . . . . . . 8 (b_|_ ^ a_|_) C (a v b_|_)
15	13	comcom7 442	. . . . . . . . 9 (b_|_ ^ a_|_) C b
16	12, 15	com2or 465	. . . . . . . 8 (b_|_ ^ a_|_) C (a v b)
17	14, 16	com2an 466	. . . . . . 7 (b_|_ ^ a_|_) C ((a v b_|_) ^ (a v b))
18	17	comcom 435	. . . . . 6 ((a v b_|_) ^ (a v b)) C (b_|_ ^ a_|_)
19	10, 18	com2or 465	. . . . 5 ((a v b_|_) ^ (a v b)) C ((b_|_ ^ a) v (b_|_ ^ a_|_))
20		comorr2 445	. . . . . . . . 9 b_|_ C (a v b_|_)
21	20	comcom6 441	. . . . . . . 8 b C (a v b_|_)
22		comorr2 445	. . . . . . . 8 b C (a v b)
23	21, 22	com2an 466	. . . . . . 7 b C ((a v b_|_) ^ (a v b))
24	23	comcom 435	. . . . . 6 ((a v b_|_) ^ (a v b)) C b
25		comor2 444	. . . . . . . . 9 (b_|_ v a) C a
26		comor1 443	. . . . . . . . 9 (b_|_ v a) C b_|_
27	25, 26	com2or 465	. . . . . . . 8 (b_|_ v a) C (a v b_|_)
28	26	comcom7 442	. . . . . . . . 9 (b_|_ v a) C b
29	25, 28	com2or 465	. . . . . . . 8 (b_|_ v a) C (a v b)
30	27, 29	com2an 466	. . . . . . 7 (b_|_ v a) C ((a v b_|_) ^ (a v b))
31	30	comcom 435	. . . . . 6 ((a v b_|_) ^ (a v b)) C (b_|_ v a)
32	24, 31	com2an 466	. . . . 5 ((a v b_|_) ^ (a v b)) C (b ^ (b_|_ v a))
33	19, 32	com2or 465	. . . 4 ((a v b_|_) ^ (a v b)) C (((b_|_ ^ a) v (b_|_ ^ a_|_)) v (b ^ (b_|_ v a)))
34		comorr 176	. . . . . . . 8 a C (a v b_|_)
35	34	comcom3 436	. . . . . . 7 a_|_ C (a v b_|_)
36		comorr 176	. . . . . . . 8 a C (a v b)
37	36	comcom3 436	. . . . . . 7 a_|_ C (a v b)
38	35, 37	com2an 466	. . . . . 6 a_|_ C ((a v b_|_) ^ (a v b))
39	38	comcom 435	. . . . 5 ((a v b_|_) ^ (a v b)) C a_|_
40		coman1 177	. . . . . . . 8 (a ^ b_|_) C a
41		coman2 178	. . . . . . . 8 (a ^ b_|_) C b_|_
42	40, 41	com2or 465	. . . . . . 7 (a ^ b_|_) C (a v b_|_)
43	41	comcom7 442	. . . . . . . 8 (a ^ b_|_) C b
44	40, 43	com2or 465	. . . . . . 7 (a ^ b_|_) C (a v b)
45	42, 44	com2an 466	. . . . . 6 (a ^ b_|_) C ((a v b_|_) ^ (a v b))
46	45	comcom 435	. . . . 5 ((a v b_|_) ^ (a v b)) C (a ^ b_|_)
47	39, 46	com2or 465	. . . 4 ((a v b_|_) ^ (a v b)) C (a_|_ v (a ^ b_|_))
48	33, 47	fh4r 458	. . 3 ((((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_))) v (((b_|_ ^ a) v (b_|_ ^ a_|_)) v (b ^ (b_|_ v a)))) = ((((a v b_|_) ^ (a v b)) v (((b_|_ ^ a) v (b_|_ ^ a_|_)) v (b ^ (b_|_ v a)))) ^ ((a_|_ v (a ^ b_|_)) v (((b_|_ ^ a) v (b_|_ ^ a_|_)) v (b ^ (b_|_ v a)))))
49		comor2 444	. . . . . . . . . 10 (a v b_|_) C b_|_
50		comor1 443	. . . . . . . . . 10 (a v b_|_) C a
51	49, 50	com2an 466	. . . . . . . . 9 (a v b_|_) C (b_|_ ^ a)
52	50	comcom2 175	. . . . . . . . . 10 (a v b_|_) C a_|_
53	49, 52	com2an 466	. . . . . . . . 9 (a v b_|_) C (b_|_ ^ a_|_)
54	51, 53	com2or 465	. . . . . . . 8 (a v b_|_) C ((b_|_ ^ a) v (b_|_ ^ a_|_))
55	49	comcom7 442	. . . . . . . . 9 (a v b_|_) C b
56	49, 50	com2or 465	. . . . . . . . 9 (a v b_|_) C (b_|_ v a)
57	55, 56	com2an 466	. . . . . . . 8 (a v b_|_) C (b ^ (b_|_ v a))
58	54, 57	com2or 465	. . . . . . 7 (a v b_|_) C (((b_|_ ^ a) v (b_|_ ^ a_|_)) v (b ^ (b_|_ v a)))
59	50, 55	com2or 465	. . . . . . 7 (a v b_|_) C (a v b)
60	58, 59	fh4r 458	. . . . . 6 (((a v b_|_) ^ (a v b)) v (((b_|_ ^ a) v (b_|_ ^ a_|_)) v (b ^ (b_|_ v a)))) = (((a v b_|_) v (((b_|_ ^ a) v (b_|_ ^ a_|_)) v (b ^ (b_|_ v a)))) ^ ((a v b) v (((b_|_ ^ a) v (b_|_ ^ a_|_)) v (b ^ (b_|_ v a)))))
61		ax-a2 30	. . . . . . . . 9 ((a v b_|_) v (((b_|_ ^ a) v (b_|_ ^ a_|_)) v (b ^ (b_|_ v a)))) = ((((b_|_ ^ a) v (b_|_ ^ a_|_)) v (b ^ (b_|_ v a))) v (a v b_|_))
62		lea 152	. . . . . . . . . . . . 13 (b_|_ ^ a) =< b_|_
63		lea 152	. . . . . . . . . . . . 13 (b_|_ ^ a_|_) =< b_|_
64	62, 63	lel2or 162	. . . . . . . . . . . 12 ((b_|_ ^ a) v (b_|_ ^ a_|_)) =< b_|_
65		leor 151	. . . . . . . . . . . 12 b_|_ =< (a v b_|_)
66	64, 65	letr 129	. . . . . . . . . . 11 ((b_|_ ^ a) v (b_|_ ^ a_|_)) =< (a v b_|_)
67		lear 153	. . . . . . . . . . . 12 (b ^ (b_|_ v a)) =< (b_|_ v a)
68		ax-a2 30	. . . . . . . . . . . 12 (b_|_ v a) = (a v b_|_)
69	67, 68	lbtr 131	. . . . . . . . . . 11 (b ^ (b_|_ v a)) =< (a v b_|_)
70	66, 69	lel2or 162	. . . . . . . . . 10 (((b_|_ ^ a) v (b_|_ ^ a_|_)) v (b ^ (b_|_ v a))) =< (a v b_|_)
71	70	df-le2 123	. . . . . . . . 9 ((((b_|_ ^ a) v (b_|_ ^ a_|_)) v (b ^ (b_|_ v a))) v (a v b_|_)) = (a v b_|_)
72	61, 71	ax-r2 35	. . . . . . . 8 ((a v b_|_) v (((b_|_ ^ a) v (b_|_ ^ a_|_)) v (b ^ (b_|_ v a)))) = (a v b_|_)
73		or12 73	. . . . . . . . 9 ((a v b) v (((b_|_ ^ a) v (b_|_ ^ a_|_)) v (b ^ (b_|_ v a)))) = (((b_|_ ^ a) v (b_|_ ^ a_|_)) v ((a v b) v (b ^ (b_|_ v a))))
74		ax-a3 31	. . . . . . . . . . 11 ((((b_|_ ^ a) v (b_|_ ^ a_|_)) v (a v b)) v (b ^ (b_|_ v a))) = (((b_|_ ^ a) v (b_|_ ^ a_|_)) v ((a v b) v (b ^ (b_|_ v a))))
75	74	ax-r1 34	. . . . . . . . . 10 (((b_|_ ^ a) v (b_|_ ^ a_|_)) v ((a v b) v (b ^ (b_|_ v a)))) = ((((b_|_ ^ a) v (b_|_ ^ a_|_)) v (a v b)) v (b ^ (b_|_ v a)))
76		ax-a3 31	. . . . . . . . . . . . 13 (((b_|_ ^ a) v (b_|_ ^ a_|_)) v (a v b)) = ((b_|_ ^ a) v ((b_|_ ^ a_|_) v (a v b)))
77		ancom 68	. . . . . . . . . . . . . . . . 17 (b_|_ ^ a_|_) = (a_|_ ^ b_|_)
78		oran 79	. . . . . . . . . . . . . . . . 17 (a v b) = (a_|_ ^ b_|_)_|_
79	77, 78	2or 67	. . . . . . . . . . . . . . . 16 ((b_|_ ^ a_|_) v (a v b)) = ((a_|_ ^ b_|_) v (a_|_ ^ b_|_)_|_)
80		df-t 40	. . . . . . . . . . . . . . . . 17 1 = ((a_|_ ^ b_|_) v (a_|_ ^ b_|_)_|_)
81	80	ax-r1 34	. . . . . . . . . . . . . . . 16 ((a_|_ ^ b_|_) v (a_|_ ^ b_|_)_|_) = 1
82	79, 81	ax-r2 35	. . . . . . . . . . . . . . 15 ((b_|_ ^ a_|_) v (a v b)) = 1
83	82	lor 66	. . . . . . . . . . . . . 14 ((b_|_ ^ a) v ((b_|_ ^ a_|_) v (a v b))) = ((b_|_ ^ a) v 1)
84		or1 96	. . . . . . . . . . . . . 14 ((b_|_ ^ a) v 1) = 1
85	83, 84	ax-r2 35	. . . . . . . . . . . . 13 ((b_|_ ^ a) v ((b_|_