Theorem u2lem1 717

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u2lem1	((a ->2 b) ->2 a) = a

Proof of Theorem u2lem1

Step	Hyp	Ref	Expression
1		df-i2 44	. 2 ((a ->2 b) ->2 a) = (a v ((a ->2 b)_|_ ^ a_|_))
2		ud2lem0c 270	. . . . . 6 (a ->2 b)_|_ = (b_|_ ^ (a v b))
3	2	ran 71	. . . . 5 ((a ->2 b)_|_ ^ a_|_) = ((b_|_ ^ (a v b)) ^ a_|_)
4		an32 76	. . . . . 6 ((b_|_ ^ (a v b)) ^ a_|_) = ((b_|_ ^ a_|_) ^ (a v b))
5		ax-a2 30	. . . . . . . . 9 (a v b) = (b v a)
6		oran 79	. . . . . . . . 9 (b v a) = (b_|_ ^ a_|_)_|_
7	5, 6	ax-r2 35	. . . . . . . 8 (a v b) = (b_|_ ^ a_|_)_|_
8	7	lan 70	. . . . . . 7 ((b_|_ ^ a_|_) ^ (a v b)) = ((b_|_ ^ a_|_) ^ (b_|_ ^ a_|_)_|_)
9		dff 93	. . . . . . . 8 0 = ((b_|_ ^ a_|_) ^ (b_|_ ^ a_|_)_|_)
10	9	ax-r1 34	. . . . . . 7 ((b_|_ ^ a_|_) ^ (b_|_ ^ a_|_)_|_) = 0
11	8, 10	ax-r2 35	. . . . . 6 ((b_|_ ^ a_|_) ^ (a v b)) = 0
12	4, 11	ax-r2 35	. . . . 5 ((b_|_ ^ (a v b)) ^ a_|_) = 0
13	3, 12	ax-r2 35	. . . 4 ((a ->2 b)_|_ ^ a_|_) = 0
14	13	lor 66	. . 3 (a v ((a ->2 b)_|_ ^ a_|_)) = (a v 0)
15		or0 94	. . 3 (a v 0) = a
16	14, 15	ax-r2 35	. 2 (a v ((a ->2 b)_|_ ^ a_|_)) = a
17	1, 16	ax-r2 35	1 ((a ->2 b) ->2 a) = a

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  0wf 10   ->2 wi2 14

This theorem is referenced by:  u2lem1n 722

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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