Theorem ud1lem3 544

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud1lem3	((a ->1 b) ->1 (a v b)) = (a v b)

Proof of Theorem ud1lem3

Step	Hyp	Ref	Expression
1		df-i1 43	. 2 ((a ->1 b) ->1 (a v b)) = ((a ->1 b)_|_ v ((a ->1 b) ^ (a v b)))
2		ud1lem0c 269	. . . 4 (a ->1 b)_|_ = (a ^ (a_|_ v b_|_))
3	2	con3 65	. . . . 5 (a ->1 b) = (a ^ (a_|_ v b_|_))_|_
4	3	ran 71	. . . 4 ((a ->1 b) ^ (a v b)) = ((a ^ (a_|_ v b_|_))_|_ ^ (a v b))
5	2, 4	2or 67	. . 3 ((a ->1 b)_|_ v ((a ->1 b) ^ (a v b))) = ((a ^ (a_|_ v b_|_)) v ((a ^ (a_|_ v b_|_))_|_ ^ (a v b)))
6		comid 179	. . . . . 6 (a ^ (a_|_ v b_|_)) C (a ^ (a_|_ v b_|_))
7	6	comcom2 175	. . . . 5 (a ^ (a_|_ v b_|_)) C (a ^ (a_|_ v b_|_))_|_
8		comor1 443	. . . . . . 7 (a v b) C a
9	8	comcom2 175	. . . . . . . 8 (a v b) C a_|_
10		comor2 444	. . . . . . . . 9 (a v b) C b
11	10	comcom2 175	. . . . . . . 8 (a v b) C b_|_
12	9, 11	com2or 465	. . . . . . 7 (a v b) C (a_|_ v b_|_)
13	8, 12	com2an 466	. . . . . 6 (a v b) C (a ^ (a_|_ v b_|_))
14	13	comcom 435	. . . . 5 (a ^ (a_|_ v b_|_)) C (a v b)
15	7, 14	fh3 453	. . . 4 ((a ^ (a_|_ v b_|_)) v ((a ^ (a_|_ v b_|_))_|_ ^ (a v b))) = (((a ^ (a_|_ v b_|_)) v (a ^ (a_|_ v b_|_))_|_) ^ ((a ^ (a_|_ v b_|_)) v (a v b)))
16		ancom 68	. . . . 5 (((a ^ (a_|_ v b_|_)) v (a ^ (a_|_ v b_|_))_|_) ^ ((a ^ (a_|_ v b_|_)) v (a v b))) = (((a ^ (a_|_ v b_|_)) v (a v b)) ^ ((a ^ (a_|_ v b_|_)) v (a ^ (a_|_ v b_|_))_|_))
17		df-t 40	. . . . . . . 8 1 = ((a ^ (a_|_ v b_|_)) v (a ^ (a_|_ v b_|_))_|_)
18	17	ax-r1 34	. . . . . . 7 ((a ^ (a_|_ v b_|_)) v (a ^ (a_|_ v b_|_))_|_) = 1
19	18	lan 70	. . . . . 6 (((a ^ (a_|_ v b_|_)) v (a v b)) ^ ((a ^ (a_|_ v b_|_)) v (a ^ (a_|_ v b_|_))_|_)) = (((a ^ (a_|_ v b_|_)) v (a v b)) ^ 1)
20		an1 98	. . . . . . 7 (((a ^ (a_|_ v b_|_)) v (a v b)) ^ 1) = ((a ^ (a_|_ v b_|_)) v (a v b))
21		comorr 176	. . . . . . . . 9 a C (a v b)
22		comorr 176	. . . . . . . . . . 11 a_|_ C (a_|_ v b_|_)
23	22	comcom2 175	. . . . . . . . . 10 a_|_ C (a_|_ v b_|_)_|_
24	23	comcom5 440	. . . . . . . . 9 a C (a_|_ v b_|_)
25	21, 24	fh4r 458	. . . . . . . 8 ((a ^ (a_|_ v b_|_)) v (a v b)) = ((a v (a v b)) ^ ((a_|_ v b_|_) v (a v b)))
26		ax-a2 30	. . . . . . . . . . 11 ((a_|_ v b_|_) v (a v b)) = ((a v b) v (a_|_ v b_|_))
27		or4 77	. . . . . . . . . . . 12 ((a v b) v (a_|_ v b_|_)) = ((a v a_|_) v (b v b_|_))
28		df-t 40	. . . . . . . . . . . . . . 15 1 = (b v b_|_)
29	28	ax-r1 34	. . . . . . . . . . . . . 14 (b v b_|_) = 1
30	29	lor 66	. . . . . . . . . . . . 13 ((a v a_|_) v (b v b_|_)) = ((a v a_|_) v 1)
31		or1 96	. . . . . . . . . . . . 13 ((a v a_|_) v 1) = 1
32	30, 31	ax-r2 35	. . . . . . . . . . . 12 ((a v a_|_) v (b v b_|_)) = 1
33	27, 32	ax-r2 35	. . . . . . . . . . 11 ((a v b) v (a_|_ v b_|_)) = 1
34	26, 33	ax-r2 35	. . . . . . . . . 10 ((a_|_ v b_|_) v (a v b)) = 1
35	34	lan 70	. . . . . . . . 9 ((a v (a v b)) ^ ((a_|_ v b_|_) v (a v b))) = ((a v (a v b)) ^ 1)
36		an1 98	. . . . . . . . . 10 ((a v (a v b)) ^ 1) = (a v (a v b))
37		ax-a3 31	. . . . . . . . . . . 12 ((a v a) v b) = (a v (a v b))
38	37	ax-r1 34	. . . . . . . . . . 11 (a v (a v b)) = ((a v a) v b)
39		oridm 102	. . . . . . . . . . . 12 (a v a) = a
40	39	ax-r5 37	. . . . . . . . . . 11 ((a v a) v b) = (a v b)
41	38, 40	ax-r2 35	. . . . . . . . . 10 (a v (a v b)) = (a v b)
42	36, 41	ax-r2 35	. . . . . . . . 9 ((a v (a v b)) ^ 1) = (a v b)
43	35, 42	ax-r2 35	. . . . . . . 8 ((a v (a v b)) ^ ((a_|_ v b_|_) v (a v b))) = (a v b)
44	25, 43	ax-r2 35	. . . . . . 7 ((a ^ (a_|_ v b_|_)) v (a v b)) = (a v b)
45	20, 44	ax-r2 35	. . . . . 6 (((a ^ (a_|_ v b_|_)) v (a v b)) ^ 1) = (a v b)
46	19, 45	ax-r2 35	. . . . 5 (((a ^ (a_|_ v b_|_)) v (a v b)) ^ ((a ^ (a_|_ v b_|_)) v (a ^ (a_|_ v b_|_))_|_)) = (a v b)
47	16, 46	ax-r2 35	. . . 4 (((a ^ (a_|_ v b_|_)) v (a ^ (a_|_ v b_|_))_|_) ^ ((a ^ (a_|_ v b_|_)) v (a v b))) = (a v b)
48	15, 47	ax-r2 35	. . 3 ((a ^ (a_|_ v b_|_)) v ((a ^ (a_|_ v b_|_))_|_ ^ (a v b))) = (a v b)
49	5, 48	ax-r2 35	. 2 ((a ->1 b)_|_ v ((a ->1 b) ^ (a v b))) = (a v b)
50	1, 49	ax-r2 35	1 ((a ->1 b) ->1 (a v b)) = (a v 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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