Theorem ud3lem1a 548

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud3lem1a	((a ->3 b)_|_ ^ (b ->3 a)) = (a ^ b_|_)

Proof of Theorem ud3lem1a

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	2an 72	. 2 ((a ->3 b)_|_ ^ (b ->3 a)) = ((((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_))) ^ (((b_|_ ^ a) v (b_|_ ^ a_|_)) v (b ^ (b_|_ v a))))
4		comor2 444	. . . . . . . . 9 (a v b_|_) C b_|_
5		comor1 443	. . . . . . . . 9 (a v b_|_) C a
6	4, 5	com2an 466	. . . . . . . 8 (a v b_|_) C (b_|_ ^ a)
7	5	comcom2 175	. . . . . . . . 9 (a v b_|_) C a_|_
8	4, 7	com2an 466	. . . . . . . 8 (a v b_|_) C (b_|_ ^ a_|_)
9	6, 8	com2or 465	. . . . . . 7 (a v b_|_) C ((b_|_ ^ a) v (b_|_ ^ a_|_))
10	9	comcom 435	. . . . . 6 ((b_|_ ^ a) v (b_|_ ^ a_|_)) C (a v b_|_)
11		comor2 444	. . . . . . . . . 10 (a v b) C b
12	11	comcom2 175	. . . . . . . . 9 (a v b) C b_|_
13		comor1 443	. . . . . . . . 9 (a v b) C a
14	12, 13	com2an 466	. . . . . . . 8 (a v b) C (b_|_ ^ a)
15	13	comcom2 175	. . . . . . . . 9 (a v b) C a_|_
16	12, 15	com2an 466	. . . . . . . 8 (a v b) C (b_|_ ^ a_|_)
17	14, 16	com2or 465	. . . . . . 7 (a v b) C ((b_|_ ^ a) v (b_|_ ^ a_|_))
18	17	comcom 435	. . . . . 6 ((b_|_ ^ a) v (b_|_ ^ a_|_)) C (a v b)
19	10, 18	com2an 466	. . . . 5 ((b_|_ ^ a) v (b_|_ ^ a_|_)) C ((a v b_|_) ^ (a v b))
20		comanr2 447	. . . . . . . . 9 a C (b_|_ ^ a)
21	20	comcom3 436	. . . . . . . 8 a_|_ C (b_|_ ^ a)
22		comanr2 447	. . . . . . . 8 a_|_ C (b_|_ ^ a_|_)
23	21, 22	com2or 465	. . . . . . 7 a_|_ C ((b_|_ ^ a) v (b_|_ ^ a_|_))
24	23	comcom 435	. . . . . 6 ((b_|_ ^ a) v (b_|_ ^ a_|_)) C a_|_
25		coman2 178	. . . . . . . . 9 (a ^ b_|_) C b_|_
26		coman1 177	. . . . . . . . 9 (a ^ b_|_) C a
27	25, 26	com2an 466	. . . . . . . 8 (a ^ b_|_) C (b_|_ ^ a)
28	26	comcom2 175	. . . . . . . . 9 (a ^ b_|_) C a_|_
29	25, 28	com2an 466	. . . . . . . 8 (a ^ b_|_) C (b_|_ ^ a_|_)
30	27, 29	com2or 465	. . . . . . 7 (a ^ b_|_) C ((b_|_ ^ a) v (b_|_ ^ a_|_))
31	30	comcom 435	. . . . . 6 ((b_|_ ^ a) v (b_|_ ^ a_|_)) C (a ^ b_|_)
32	24, 31	com2or 465	. . . . 5 ((b_|_ ^ a) v (b_|_ ^ a_|_)) C (a_|_ v (a ^ b_|_))
33	19, 32	com2an 466	. . . 4 ((b_|_ ^ a) v (b_|_ ^ a_|_)) C (((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_)))
34		comanr1 446	. . . . . . . 8 b_|_ C (b_|_ ^ a)
35	34	comcom6 441	. . . . . . 7 b C (b_|_ ^ a)
36		comanr1 446	. . . . . . . 8 b_|_ C (b_|_ ^ a_|_)
37	36	comcom6 441	. . . . . . 7 b C (b_|_ ^ a_|_)
38	35, 37	com2or 465	. . . . . 6 b C ((b_|_ ^ a) v (b_|_ ^ a_|_))
39	38	comcom 435	. . . . 5 ((b_|_ ^ a) v (b_|_ ^ a_|_)) C b
40		comor1 443	. . . . . . . 8 (b_|_ v a) C b_|_
41		comor2 444	. . . . . . . 8 (b_|_ v a) C a
42	40, 41	com2an 466	. . . . . . 7 (b_|_ v a) C (b_|_ ^ a)
43	41	comcom2 175	. . . . . . . 8 (b_|_ v a) C a_|_
44	40, 43	com2an 466	. . . . . . 7 (b_|_ v a) C (b_|_ ^ a_|_)
45	42, 44	com2or 465	. . . . . 6 (b_|_ v a) C ((b_|_ ^ a) v (b_|_ ^ a_|_))
46	45	comcom 435	. . . . 5 ((b_|_ ^ a) v (b_|_ ^ a_|_)) C (b_|_ v a)
47	39, 46	com2an 466	. . . 4 ((b_|_ ^ a) v (b_|_ ^ a_|_)) C (b ^ (b_|_ v a))
48	33, 47	fh2 452	. . 3 ((((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_))) ^ (((b_|_ ^ a) v (b_|_ ^ a_|_)) v (b ^ (b_|_ v a)))) = (((((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_))) ^ ((b_|_ ^ a) v (b_|_ ^ a_|_))) v ((((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_))) ^ (b ^ (b_|_ v a))))
49		coman2 178	. . . . . . . . . 10 (b_|_ ^ a) C a
50		coman1 177	. . . . . . . . . 10 (b_|_ ^ a) C b_|_
51	49, 50	com2or 465	. . . . . . . . 9 (b_|_ ^ a) C (a v b_|_)
52	50	comcom7 442	. . . . . . . . . 10 (b_|_ ^ a) C b
53	49, 52	com2or 465	. . . . . . . . 9 (b_|_ ^ a) C (a v b)
54	51, 53	com2an 466	. . . . . . . 8 (b_|_ ^ a) C ((a v b_|_) ^ (a v b))
55	49	comcom2 175	. . . . . . . . 9 (b_|_ ^ a) C a_|_
56	49, 50	com2an 466	. . . . . . . . 9 (b_|_ ^ a) C (a ^ b_|_)
57	55, 56	com2or 465	. . . . . . . 8 (b_|_ ^ a) C (a_|_ v (a ^ b_|_))
58	54, 57	com2an 466	. . . . . . 7 (b_|_ ^ a) C (((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_)))
59	50, 55	com2an 466	. . . . . . 7 (b_|_ ^ a) C (b_|_ ^ a_|_)
60	58, 59	fh2 452	. . . . . 6 ((((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_))) ^ ((b_|_ ^ a) v (b_|_ ^ a_|_))) = (((((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_))) ^ (b_|_ ^ a)) v ((((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_))) ^ (b_|_ ^ a_|_)))
61		anass 69	. . . . . . . . 9 ((((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_))) ^ (b_|_ ^ a)) = (((a v b_|_) ^ (a v b)) ^ ((a_|_ v (a ^ b_|_)) ^ (b_|_ ^ a)))
62		ancom 68	. . . . . . . . . . . 12 ((a_|_ v (a ^ b_|_)) ^ (b_|_ ^ a)) = ((b_|_ ^ a) ^ (a_|_ v (a ^ b_|_)))
63		ancom 68	. . . . . . . . . . . . . 14 (b_|_ ^ a) = (a ^ b_|_)
64		ax-a2 30	. . . . . . . . . . . . . 14 (a_|_ v (a ^ b_|_)) = ((a ^ b_|_) v a_|_)
65	63, 64	2an 72	. . . . . . . . . . . . 13 ((b_|_ ^ a) ^ (a_|_ v (a ^ b_|_))) = ((a ^ b_|_) ^ ((a ^ b_|_) v a_|_))
66		a5c 113	. . . . . . . . . . . . 13 ((a ^ b_|_) ^ ((a ^ b_|_) v a_|_)) = (a ^ b_|_)
67	65, 66	ax-r2 35	. . . . . . . . . . . 12 ((b_|_ ^ a) ^ (a_|_ v (a ^ b_|_))) = (a ^ b_|_)
68	62, 67	ax-r2 35	. . . . . . . . . . 11 ((a_|_ v (a ^ b_|_)) ^ (b_|_ ^ a)) = (a ^ b_|_)
69	68	lan 70	. . . . . . . . . 10 (((a v b_|_) ^ (a v b)) ^ ((a_|_ v (a ^ b_|_)) ^ (b_|_ ^ a))) = (((a v b_|_) ^ (a v b)) ^ (a ^ b_|_))
70		ancom 68	. . . . . . . . . . 11 (((a v b_|_) ^ (a v b)) ^ (a ^ b_|_)) = ((a ^ b_|_) ^ ((a v b_|_) ^ (a v b)))
71		lea 152	. . . . . . . . . . . . 13 (a ^ b_|_) =< a
72		leo 150	. . . . . . . . . . . . . 14 a =< (a v b_|_)
73		leo 150	. . . . . . . . . . . . . 14 a =< (a v b)
74	72, 73	ler2an 165	. . . . . . . . . . . . 13 a =< ((a v b_|_) ^ (a v b))
75	71, 74	letr 129	. . . . . . . . . . . 12 (a ^ b_|_) =< ((a v b_|_) ^ (a v b))
76	75	df2le2 128	. . . . . . . . . . 11 ((a ^ b_|_) ^ ((a v b_|_) ^ (a v b))) = (a ^ b_|_)
77	70, 76	ax-r2 35	. . . . . . . . . 10 (((a v b_|_) ^ (a v b)) ^ (a ^ b_|_)) = (a ^ b_|_)
78	69, 77	ax-r2 35	. . . . . . . . 9 (((a v b_|_) ^ (a v b)) ^ ((a_|_ v (a ^ b_|_)) ^ (b_|_ ^ a))) = (a ^ b_|_)
79	61, 78	ax-r2 35	. . . . . . . 8 ((((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_))) ^ (b_|_ ^ a)) = (a ^ b_|_)
80		an32 76	. . . . . . . . 9 ((((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_))) ^ (b_|_ ^ a_|_)) = ((((a v b_|_) ^ (a v b)) ^ (b_|_ ^ a_|_)) ^ (a_|_ v (a ^ b_|_)))
81		anass 69	. . . . . . . . . . . 12 (((a v b_|_) ^ (a v b)) ^ (b_|_ ^ a_|_)) = ((a v b_|_) ^ ((a v b) ^ (b_|_ ^ a_|_)))
82		ax-a2 30	. . . . . . . . . . . . . . . 16 (a v b) = (b v a)
83		oran 79	. . . . . . . . . . . . . . . . . 18 (b v a) = (b_|_ ^ a_|_)_|_
84	83	ax-r1 34	. . . . . . . . . . . . . . . . 17 (b_|_ ^ a_|_