Theorem u2lem7 755

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u2lem7	(a ->2 (a_|_ ->2 b)) = (((a ^ b_|_) v (a_|_ ^ b_|_)) v b)

Proof of Theorem u2lem7

Step	Hyp	Ref	Expression
1		df-i2 44	. 2 (a ->2 (a_|_ ->2 b)) = ((a_|_ ->2 b) v (a_|_ ^ (a_|_ ->2 b)_|_))
2		df-i2 44	. . . . 5 (a_|_ ->2 b) = (b v (a_|__|_ ^ b_|_))
3		ax-a1 29	. . . . . . . 8 a = a_|__|_
4	3	ax-r1 34	. . . . . . 7 a_|__|_ = a
5	4	ran 71	. . . . . 6 (a_|__|_ ^ b_|_) = (a ^ b_|_)
6	5	lor 66	. . . . 5 (b v (a_|__|_ ^ b_|_)) = (b v (a ^ b_|_))
7	2, 6	ax-r2 35	. . . 4 (a_|_ ->2 b) = (b v (a ^ b_|_))
8		ancom 68	. . . . 5 (a_|_ ^ (a_|_ ->2 b)_|_) = ((a_|_ ->2 b)_|_ ^ a_|_)
9		u2lemnaa 623	. . . . 5 ((a_|_ ->2 b)_|_ ^ a_|_) = (a_|_ ^ b_|_)
10	8, 9	ax-r2 35	. . . 4 (a_|_ ^ (a_|_ ->2 b)_|_) = (a_|_ ^ b_|_)
11	7, 10	2or 67	. . 3 ((a_|_ ->2 b) v (a_|_ ^ (a_|_ ->2 b)_|_)) = ((b v (a ^ b_|_)) v (a_|_ ^ b_|_))
12		ax-a3 31	. . . 4 ((b v (a ^ b_|_)) v (a_|_ ^ b_|_)) = (b v ((a ^ b_|_) v (a_|_ ^ b_|_)))
13		ax-a2 30	. . . 4 (b v ((a ^ b_|_) v (a_|_ ^ b_|_))) = (((a ^ b_|_) v (a_|_ ^ b_|_)) v b)
14	12, 13	ax-r2 35	. . 3 ((b v (a ^ b_|_)) v (a_|_ ^ b_|_)) = (((a ^ b_|_) v (a_|_ ^ b_|_)) v b)
15	11, 14	ax-r2 35	. 2 ((a_|_ ->2 b) v (a_|_ ^ (a_|_ ->2 b)_|_)) = (((a ^ b_|_) v (a_|_ ^ b_|_)) v b)
16	1, 15	ax-r2 35	1 (a ->2 (a_|_ ->2 b)) = (((a ^ b_|_) v (a_|_ ^ b_|_)) v b)

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

This theorem is referenced by:  u2lem7n 757  u2lem8 764

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-i2 44  df-le1 122  df-le2 123

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