Theorem u3lem3 733

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u3lem3	(a ->3 (b ->3 a)) = (a v ((a_|_ ^ b) v (a_|_ ^ b_|_)))

Proof of Theorem u3lem3

Step	Hyp	Ref	Expression
1		df-i3 45	. 2 (a ->3 (b ->3 a)) = (((a_|_ ^ (b ->3 a)) v (a_|_ ^ (b ->3 a)_|_)) v (a ^ (a_|_ v (b ->3 a))))
2		ancom 68	. . . . . . 7 (a_|_ ^ (b ->3 a)) = ((b ->3 a) ^ a_|_)
3		u3lemanb 599	. . . . . . 7 ((b ->3 a) ^ a_|_) = (b_|_ ^ a_|_)
4	2, 3	ax-r2 35	. . . . . 6 (a_|_ ^ (b ->3 a)) = (b_|_ ^ a_|_)
5		ancom 68	. . . . . . 7 (a_|_ ^ (b ->3 a)_|_) = ((b ->3 a)_|_ ^ a_|_)
6		u3lemnanb 639	. . . . . . 7 ((b ->3 a)_|_ ^ a_|_) = (b ^ a_|_)
7	5, 6	ax-r2 35	. . . . . 6 (a_|_ ^ (b ->3 a)_|_) = (b ^ a_|_)
8	4, 7	2or 67	. . . . 5 ((a_|_ ^ (b ->3 a)) v (a_|_ ^ (b ->3 a)_|_)) = ((b_|_ ^ a_|_) v (b ^ a_|_))
9		ancom 68	. . . . . . 7 (b_|_ ^ a_|_) = (a_|_ ^ b_|_)
10		ancom 68	. . . . . . 7 (b ^ a_|_) = (a_|_ ^ b)
11	9, 10	2or 67	. . . . . 6 ((b_|_ ^ a_|_) v (b ^ a_|_)) = ((a_|_ ^ b_|_) v (a_|_ ^ b))
12		ax-a2 30	. . . . . 6 ((a_|_ ^ b_|_) v (a_|_ ^ b)) = ((a_|_ ^ b) v (a_|_ ^ b_|_))
13	11, 12	ax-r2 35	. . . . 5 ((b_|_ ^ a_|_) v (b ^ a_|_)) = ((a_|_ ^ b) v (a_|_ ^ b_|_))
14	8, 13	ax-r2 35	. . . 4 ((a_|_ ^ (b ->3 a)) v (a_|_ ^ (b ->3 a)_|_)) = ((a_|_ ^ b) v (a_|_ ^ b_|_))
15		ax-a2 30	. . . . . . 7 (a_|_ v (b ->3 a)) = ((b ->3 a) v a_|_)
16		u3lemonb 619	. . . . . . 7 ((b ->3 a) v a_|_) = 1
17	15, 16	ax-r2 35	. . . . . 6 (a_|_ v (b ->3 a)) = 1
18	17	lan 70	. . . . 5 (a ^ (a_|_ v (b ->3 a))) = (a ^ 1)
19		an1 98	. . . . 5 (a ^ 1) = a
20	18, 19	ax-r2 35	. . . 4 (a ^ (a_|_ v (b ->3 a))) = a
21	14, 20	2or 67	. . 3 (((a_|_ ^ (b ->3 a)) v (a_|_ ^ (b ->3 a)_|_)) v (a ^ (a_|_ v (b ->3 a)))) = (((a_|_ ^ b) v (a_|_ ^ b_|_)) v a)
22		ax-a2 30	. . 3 (((a_|_ ^ b) v (a_|_ ^ b_|_)) v a) = (a v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
23	21, 22	ax-r2 35	. 2 (((a_|_ ^ (b ->3 a)) v (a_|_ ^ (b ->3 a)_|_)) v (a ^ (a_|_ v (b ->3 a)))) = (a v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
24	1, 23	ax-r2 35	1 (a ->3 (b ->3 a)) = (a v ((a_|_ ^ b) v (a_|_ ^ b_|_)))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->3 wi3 15

This theorem is referenced by:  u3lem3n 736  u3lem14a 773

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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