Theorem ud2lem3 547

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud2lem3	((a ->2 b) ->2 (a v b)) = (a v b)

Proof of Theorem ud2lem3

Step	Hyp	Ref	Expression
1		df-i2 44	. 2 ((a ->2 b) ->2 (a v b)) = ((a v b) v ((a ->2 b)_|_ ^ (a v b)_|_))
2		ud2lem0c 270	. . . . 5 (a ->2 b)_|_ = (b_|_ ^ (a v b))
3	2	ran 71	. . . 4 ((a ->2 b)_|_ ^ (a v b)_|_) = ((b_|_ ^ (a v b)) ^ (a v b)_|_)
4	3	lor 66	. . 3 ((a v b) v ((a ->2 b)_|_ ^ (a v b)_|_)) = ((a v b) v ((b_|_ ^ (a v b)) ^ (a v b)_|_))
5		coman2 178	. . . . . 6 (b_|_ ^ (a v b)) C (a v b)
6	5	comcom 435	. . . . 5 (a v b) C (b_|_ ^ (a v b))
7		comid 179	. . . . . 6 (a v b) C (a v b)
8	7	comcom2 175	. . . . 5 (a v b) C (a v b)_|_
9	6, 8	fh3 453	. . . 4 ((a v b) v ((b_|_ ^ (a v b)) ^ (a v b)_|_)) = (((a v b) v (b_|_ ^ (a v b))) ^ ((a v b) v (a v b)_|_))
10		ancom 68	. . . . . . 7 (b_|_ ^ (a v b)) = ((a v b) ^ b_|_)
11	10	lor 66	. . . . . 6 ((a v b) v (b_|_ ^ (a v b))) = ((a v b) v ((a v b) ^ b_|_))
12		df-t 40	. . . . . . 7 1 = ((a v b) v (a v b)_|_)
13	12	ax-r1 34	. . . . . 6 ((a v b) v (a v b)_|_) = 1
14	11, 13	2an 72	. . . . 5 (((a v b) v (b_|_ ^ (a v b))) ^ ((a v b) v (a v b)_|_)) = (((a v b) v ((a v b) ^ b_|_)) ^ 1)
15		an1 98	. . . . . 6 (((a v b) v ((a v b) ^ b_|_)) ^ 1) = ((a v b) v ((a v b) ^ b_|_))
16		a5b 112	. . . . . 6 ((a v b) v ((a v b) ^ b_|_)) = (a v b)
17	15, 16	ax-r2 35	. . . . 5 (((a v b) v ((a v b) ^ b_|_)) ^ 1) = (a v b)
18	14, 17	ax-r2 35	. . . 4 (((a v b) v (b_|_ ^ (a v b))) ^ ((a v b) v (a v b)_|_)) = (a v b)
19	9, 18	ax-r2 35	. . 3 ((a v b) v ((b_|_ ^ (a v b)) ^ (a v b)_|_)) = (a v b)
20	4, 19	ax-r2 35	. 2 ((a v b) v ((a ->2 b)_|_ ^ (a v b)_|_)) = (a v b)
21	1, 20	ax-r2 35	1 ((a ->2 b) ->2 (a v b)) = (a v b)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->2 wi2 14

This theorem is referenced by:  ud2 578

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-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

metamath.org