Theorem u3lem3n 736

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u3lem3n	(a ->3 (b ->3 a))_|_ = (a_|_ ^ ((a v b) ^ (a v b_|_)))

Proof of Theorem u3lem3n

Step	Hyp	Ref	Expression
1		u3lem3 733	. . 3 (a ->3 (b ->3 a)) = (a v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
2		ax-a2 30	. . . . . 6 ((a_|_ ^ b) v (a_|_ ^ b_|_)) = ((a_|_ ^ b_|_) v (a_|_ ^ b))
3		anor3 82	. . . . . . . 8 (a_|_ ^ b_|_) = (a v b)_|_
4		anor2 81	. . . . . . . 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		oran3 85	. . . . . . 7 ((a v b)_|_ v (a v b_|_)_|_) = ((a v b) ^ (a v b_|_))_|_
7	5, 6	ax-r2 35	. . . . . 6 ((a_|_ ^ b_|_) v (a_|_ ^ b)) = ((a v b) ^ (a v b_|_))_|_
8	2, 7	ax-r2 35	. . . . 5 ((a_|_ ^ b) v (a_|_ ^ b_|_)) = ((a v b) ^ (a v b_|_))_|_
9	8	lor 66	. . . 4 (a v ((a_|_ ^ b) v (a_|_ ^ b_|_))) = (a v ((a v b) ^ (a v b_|_))_|_)
10		oran1 83	. . . 4 (a v ((a v b) ^ (a v b_|_))_|_) = (a_|_ ^ ((a v b) ^ (a v b_|_)))_|_
11	9, 10	ax-r2 35	. . 3 (a v ((a_|_ ^ b) v (a_|_ ^ b_|_))) = (a_|_ ^ ((a v b) ^ (a v b_|_)))_|_
12	1, 11	ax-r2 35	. 2 (a ->3 (b ->3 a)) = (a_|_ ^ ((a v b) ^ (a v b_|_)))_|_
13	12	con2 64	1 (a ->3 (b ->3 a))_|_ = (a_|_ ^ ((a v b) ^ (a v b_|_)))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->3 wi3 15

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-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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