Theorem ud1lem1 542

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud1lem1	((a ->1 b) ->1 (b ->1 a)) = (a v (a_|_ ^ b_|_))

Proof of Theorem ud1lem1

Step	Hyp	Ref	Expression
1		df-i1 43	. 2 ((a ->1 b) ->1 (b ->1 a)) = ((a ->1 b)_|_ v ((a ->1 b) ^ (b ->1 a)))
2		ud1lem0c 269	. . . 4 (a ->1 b)_|_ = (a ^ (a_|_ v b_|_))
3		df-i1 43	. . . . 5 (a ->1 b) = (a_|_ v (a ^ b))
4		df-i1 43	. . . . 5 (b ->1 a) = (b_|_ v (b ^ a))
5	3, 4	2an 72	. . . 4 ((a ->1 b) ^ (b ->1 a)) = ((a_|_ v (a ^ b)) ^ (b_|_ v (b ^ a)))
6	2, 5	2or 67	. . 3 ((a ->1 b)_|_ v ((a ->1 b) ^ (b ->1 a))) = ((a ^ (a_|_ v b_|_)) v ((a_|_ v (a ^ b)) ^ (b_|_ v (b ^ a))))
7		ancom 68	. . . . . . . 8 (b ^ a) = (a ^ b)
8	7	lor 66	. . . . . . 7 (b_|_ v (b ^ a)) = (b_|_ v (a ^ b))
9	8	lan 70	. . . . . 6 ((a_|_ v (a ^ b)) ^ (b_|_ v (b ^ a))) = ((a_|_ v (a ^ b)) ^ (b_|_ v (a ^ b)))
10		coman1 177	. . . . . . . . 9 (a ^ b) C a
11	10	comcom2 175	. . . . . . . 8 (a ^ b) C a_|_
12		coman2 178	. . . . . . . . 9 (a ^ b) C b
13	12	comcom2 175	. . . . . . . 8 (a ^ b) C b_|_
14	11, 13	fh3r 457	. . . . . . 7 ((a_|_ ^ b_|_) v (a ^ b)) = ((a_|_ v (a ^ b)) ^ (b_|_ v (a ^ b)))
15	14	ax-r1 34	. . . . . 6 ((a_|_ v (a ^ b)) ^ (b_|_ v (a ^ b))) = ((a_|_ ^ b_|_) v (a ^ b))
16	9, 15	ax-r2 35	. . . . 5 ((a_|_ v (a ^ b)) ^ (b_|_ v (b ^ a))) = ((a_|_ ^ b_|_) v (a ^ b))
17	16	lor 66	. . . 4 ((a ^ (a_|_ v b_|_)) v ((a_|_ v (a ^ b)) ^ (b_|_ v (b ^ a)))) = ((a ^ (a_|_ v b_|_)) v ((a_|_ ^ b_|_) v (a ^ b)))
18		or12 73	. . . . 5 ((a ^ (a_|_ v b_|_)) v ((a_|_ ^ b_|_) v (a ^ b))) = ((a_|_ ^ b_|_) v ((a ^ (a_|_ v b_|_)) v (a ^ b)))
19	10	comcom 435	. . . . . . . 8 a C (a ^ b)
20		comorr 176	. . . . . . . . . 10 a_|_ C (a_|_ v b_|_)
21	20	comcom2 175	. . . . . . . . 9 a_|_ C (a_|_ v b_|_)_|_
22	21	comcom5 440	. . . . . . . 8 a C (a_|_ v b_|_)
23	19, 22	fh4r 458	. . . . . . 7 ((a ^ (a_|_ v b_|_)) v (a ^ b)) = ((a v (a ^ b)) ^ ((a_|_ v b_|_) v (a ^ b)))
24	23	lor 66	. . . . . 6 ((a_|_ ^ b_|_) v ((a ^ (a_|_ v b_|_)) v (a ^ b))) = ((a_|_ ^ b_|_) v ((a v (a ^ b)) ^ ((a_|_ v b_|_) v (a ^ b))))
25		a5b 112	. . . . . . . . . 10 (a v (a ^ b)) = a
26		df-a 39	. . . . . . . . . . . 12 (a ^ b) = (a_|_ v b_|_)_|_
27	26	lor 66	. . . . . . . . . . 11 ((a_|_ v b_|_) v (a ^ b)) = ((a_|_ v b_|_) v (a_|_ v b_|_)_|_)
28		df-t 40	. . . . . . . . . . . 12 1 = ((a_|_ v b_|_) v (a_|_ v b_|_)_|_)
29	28	ax-r1 34	. . . . . . . . . . 11 ((a_|_ v b_|_) v (a_|_ v b_|_)_|_) = 1
30	27, 29	ax-r2 35	. . . . . . . . . 10 ((a_|_ v b_|_) v (a ^ b)) = 1
31	25, 30	2an 72	. . . . . . . . 9 ((a v (a ^ b)) ^ ((a_|_ v b_|_) v (a ^ b))) = (a ^ 1)
32		an1 98	. . . . . . . . 9 (a ^ 1) = a
33	31, 32	ax-r2 35	. . . . . . . 8 ((a v (a ^ b)) ^ ((a_|_ v b_|_) v (a ^ b))) = a
34	33	lor 66	. . . . . . 7 ((a_|_ ^ b_|_) v ((a v (a ^ b)) ^ ((a_|_ v b_|_) v (a ^ b)))) = ((a_|_ ^ b_|_) v a)
35		ax-a2 30	. . . . . . 7 ((a_|_ ^ b_|_) v a) = (a v (a_|_ ^ b_|_))
36	34, 35	ax-r2 35	. . . . . 6 ((a_|_ ^ b_|_) v ((a v (a ^ b)) ^ ((a_|_ v b_|_) v (a ^ b)))) = (a v (a_|_ ^ b_|_))
37	24, 36	ax-r2 35	. . . . 5 ((a_|_ ^ b_|_) v ((a ^ (a_|_ v b_|_)) v (a ^ b))) = (a v (a_|_ ^ b_|_))
38	18, 37	ax-r2 35	. . . 4 ((a ^ (a_|_ v b_|_)) v ((a_|_ ^ b_|_) v (a ^ b))) = (a v (a_|_ ^ b_|_))
39	17, 38	ax-r2 35	. . 3 ((a ^ (a_|_ v b_|_)) v ((a_|_ v (a ^ b)) ^ (b_|_ v (b ^ a)))) = (a v (a_|_ ^ b_|_))
40	6, 39	ax-r2 35	. 2 ((a ->1 b)_|_ v ((a ->1 b) ^ (b ->1 a))) = (a v (a_|_ ^ b_|_))
41	1, 40	ax-r2 35	1 ((a ->1 b) ->1 (b ->1 a)) = (a v (a_|_ ^ 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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