Theorem u2lem7n 757

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u2lem7n	(a ->2 (a_|_ ->2 b))_|_ = (((a v b) ^ (a_|_ v b)) ^ b_|_)

Proof of Theorem u2lem7n

Step	Hyp	Ref	Expression
1		u2lem7 755	. . 3 (a ->2 (a_|_ ->2 b)) = (((a ^ b_|_) v (a_|_ ^ b_|_)) v b)
2		ax-a2 30	. . . . . . 7 ((a ^ b_|_) v (a_|_ ^ b_|_)) = ((a_|_ ^ b_|_) v (a ^ b_|_))
3		anor3 82	. . . . . . . 8 (a_|_ ^ b_|_) = (a v b)_|_
4		anor1 80	. . . . . . . 8 (a ^ b_|_) = (a_|_ v b)_|_
5	3, 4	2or 67	. . . . . . 7 ((a_|_ ^ b_|_) v (a ^ b_|_)) = ((a v b)_|_ v (a_|_ v b)_|_)
6	2, 5	ax-r2 35	. . . . . 6 ((a ^ b_|_) v (a_|_ ^ b_|_)) = ((a v b)_|_ v (a_|_ v b)_|_)
7		oran3 85	. . . . . 6 ((a v b)_|_ v (a_|_ v b)_|_) = ((a v b) ^ (a_|_ v b))_|_
8	6, 7	ax-r2 35	. . . . 5 ((a ^ b_|_) v (a_|_ ^ b_|_)) = ((a v b) ^ (a_|_ v b))_|_
9	8	ax-r5 37	. . . 4 (((a ^ b_|_) v (a_|_ ^ b_|_)) v b) = (((a v b) ^ (a_|_ v b))_|_ v b)
10		oran2 84	. . . 4 (((a v b) ^ (a_|_ v b))_|_ v b) = (((a v b) ^ (a_|_ v b)) ^ b_|_)_|_
11	9, 10	ax-r2 35	. . 3 (((a ^ b_|_) v (a_|_ ^ b_|_)) v b) = (((a v b) ^ (a_|_ v b)) ^ b_|_)_|_
12	1, 11	ax-r2 35	. 2 (a ->2 (a_|_ ->2 b)) = (((a v b) ^ (a_|_ v b)) ^ b_|_)_|_
13	12	con2 64	1 (a ->2 (a_|_ ->2 b))_|_ = (((a v b) ^ (a_|_ v b)) ^ b_|_)

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

This theorem is referenced by:  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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