Theorem ud2lem1 545

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud2lem1	((a ->2 b) ->2 (b ->2 a)) = (a v (a_|_ ^ b_|_))

Proof of Theorem ud2lem1

Step	Hyp	Ref	Expression
1		df-i2 44	. 2 ((a ->2 b) ->2 (b ->2 a)) = ((b ->2 a) v ((a ->2 b)_|_ ^ (b ->2 a)_|_))
2		df-i2 44	. . . 4 (b ->2 a) = (a v (b_|_ ^ a_|_))
3		ud2lem0c 270	. . . . 5 (a ->2 b)_|_ = (b_|_ ^ (a v b))
4		ud2lem0c 270	. . . . 5 (b ->2 a)_|_ = (a_|_ ^ (b v a))
5	3, 4	2an 72	. . . 4 ((a ->2 b)_|_ ^ (b ->2 a)_|_) = ((b_|_ ^ (a v b)) ^ (a_|_ ^ (b v a)))
6	2, 5	2or 67	. . 3 ((b ->2 a) v ((a ->2 b)_|_ ^ (b ->2 a)_|_)) = ((a v (b_|_ ^ a_|_)) v ((b_|_ ^ (a v b)) ^ (a_|_ ^ (b v a))))
7		ancom 68	. . . . . 6 (b_|_ ^ a_|_) = (a_|_ ^ b_|_)
8	7	lor 66	. . . . 5 (a v (b_|_ ^ a_|_)) = (a v (a_|_ ^ b_|_))
9		dff 93	. . . . . . . 8 0 = ((b_|_ ^ a_|_) ^ (b_|_ ^ a_|_)_|_)
10		oran 79	. . . . . . . . . 10 (b v a) = (b_|_ ^ a_|_)_|_
11	10	ax-r1 34	. . . . . . . . 9 (b_|_ ^ a_|_)_|_ = (b v a)
12	11	lan 70	. . . . . . . 8 ((b_|_ ^ a_|_) ^ (b_|_ ^ a_|_)_|_) = ((b_|_ ^ a_|_) ^ (b v a))
13	9, 12	ax-r2 35	. . . . . . 7 0 = ((b_|_ ^ a_|_) ^ (b v a))
14		anandir 107	. . . . . . . 8 ((b_|_ ^ a_|_) ^ (b v a)) = ((b_|_ ^ (b v a)) ^ (a_|_ ^ (b v a)))
15		ax-a2 30	. . . . . . . . . 10 (b v a) = (a v b)
16	15	lan 70	. . . . . . . . 9 (b_|_ ^ (b v a)) = (b_|_ ^ (a v b))
17	16	ran 71	. . . . . . . 8 ((b_|_ ^ (b v a)) ^ (a_|_ ^ (b v a))) = ((b_|_ ^ (a v b)) ^ (a_|_ ^ (b v a)))
18	14, 17	ax-r2 35	. . . . . . 7 ((b_|_ ^ a_|_) ^ (b v a)) = ((b_|_ ^ (a v b)) ^ (a_|_ ^ (b v a)))
19	13, 18	ax-r2 35	. . . . . 6 0 = ((b_|_ ^ (a v b)) ^ (a_|_ ^ (b v a)))
20	19	ax-r1 34	. . . . 5 ((b_|_ ^ (a v b)) ^ (a_|_ ^ (b v a))) = 0
21	8, 20	2or 67	. . . 4 ((a v (b_|_ ^ a_|_)) v ((b_|_ ^ (a v b)) ^ (a_|_ ^ (b v a)))) = ((a v (a_|_ ^ b_|_)) v 0)
22		or0 94	. . . 4 ((a v (a_|_ ^ b_|_)) v 0) = (a v (a_|_ ^ b_|_))
23	21, 22	ax-r2 35	. . 3 ((a v (b_|_ ^ a_|_)) v ((b_|_ ^ (a v b)) ^ (a_|_ ^ (b v a)))) = (a v (a_|_ ^ b_|_))
24	6, 23	ax-r2 35	. 2 ((b ->2 a) v ((a ->2 b)_|_ ^ (b ->2 a)_|_)) = (a v (a_|_ ^ b_|_))
25	1, 24	ax-r2 35	1 ((a ->2 b) ->2 (b ->2 a)) = (a v (a_|_ ^ b_|_))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  0wf 10   ->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-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i2 44

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