Theorem u3lem14a 773

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u3lem14a	(a ->3 ((b ->3 a_|_) ->3 b_|_)) = (a ->3 (b ->3 a))

Proof of Theorem u3lem14a

Step	Hyp	Ref	Expression
1		u3lem13b 772	. . 3 ((b ->3 a_|_) ->3 b_|_) = (b ->1 a)
2	1	ud3lem0a 252	. 2 (a ->3 ((b ->3 a_|_) ->3 b_|_)) = (a ->3 (b ->1 a))
3		df-i3 45	. . 3 (a ->3 (b ->1 a)) = (((a_|_ ^ (b ->1 a)) v (a_|_ ^ (b ->1 a)_|_)) v (a ^ (a_|_ v (b ->1 a))))
4		ancom 68	. . . . . . . 8 (a_|_ ^ (b ->1 a)) = ((b ->1 a) ^ a_|_)
5		u1lemanb 597	. . . . . . . 8 ((b ->1 a) ^ a_|_) = (b_|_ ^ a_|_)
6	4, 5	ax-r2 35	. . . . . . 7 (a_|_ ^ (b ->1 a)) = (b_|_ ^ a_|_)
7		ancom 68	. . . . . . . 8 (a_|_ ^ (b ->1 a)_|_) = ((b ->1 a)_|_ ^ a_|_)
8		u1lemnanb 637	. . . . . . . 8 ((b ->1 a)_|_ ^ a_|_) = (b ^ a_|_)
9	7, 8	ax-r2 35	. . . . . . 7 (a_|_ ^ (b ->1 a)_|_) = (b ^ a_|_)
10	6, 9	2or 67	. . . . . 6 ((a_|_ ^ (b ->1 a)) v (a_|_ ^ (b ->1 a)_|_)) = ((b_|_ ^ a_|_) v (b ^ a_|_))
11		ax-a2 30	. . . . . . 7 ((b_|_ ^ a_|_) v (b ^ a_|_)) = ((b ^ a_|_) v (b_|_ ^ a_|_))
12		ancom 68	. . . . . . . 8 (b ^ a_|_) = (a_|_ ^ b)
13		ancom 68	. . . . . . . 8 (b_|_ ^ a_|_) = (a_|_ ^ b_|_)
14	12, 13	2or 67	. . . . . . 7 ((b ^ a_|_) v (b_|_ ^ a_|_)) = ((a_|_ ^ b) v (a_|_ ^ b_|_))
15	11, 14	ax-r2 35	. . . . . 6 ((b_|_ ^ a_|_) v (b ^ a_|_)) = ((a_|_ ^ b) v (a_|_ ^ b_|_))
16	10, 15	ax-r2 35	. . . . 5 ((a_|_ ^ (b ->1 a)) v (a_|_ ^ (b ->1 a)_|_)) = ((a_|_ ^ b) v (a_|_ ^ b_|_))
17		ax-a2 30	. . . . . . . 8 (a_|_ v (b ->1 a)) = ((b ->1 a) v a_|_)
18		u1lemonb 617	. . . . . . . 8 ((b ->1 a) v a_|_) = 1
19	17, 18	ax-r2 35	. . . . . . 7 (a_|_ v (b ->1 a)) = 1
20	19	lan 70	. . . . . 6 (a ^ (a_|_ v (b ->1 a))) = (a ^ 1)
21		an1 98	. . . . . 6 (a ^ 1) = a
22	20, 21	ax-r2 35	. . . . 5 (a ^ (a_|_ v (b ->1 a))) = a
23	16, 22	2or 67	. . . 4 (((a_|_ ^ (b ->1 a)) v (a_|_ ^ (b ->1 a)_|_)) v (a ^ (a_|_ v (b ->1 a)))) = (((a_|_ ^ b) v (a_|_ ^ b_|_)) v a)
24		ax-a2 30	. . . . 5 (((a_|_ ^ b) v (a_|_ ^ b_|_)) v a) = (a v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
25		u3lem3 733	. . . . . . 7 (a ->3 (b ->3 a)) = (a v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
26	25	ax-r1 34	. . . . . 6 (a v ((a_|_ ^ b) v (a_|_ ^ b_|_))) = (a ->3 (b ->3 a))
27		id 58	. . . . . 6 (a ->3 (b ->3 a)) = (a ->3 (b ->3 a))
28	26, 27	ax-r2 35	. . . . 5 (a v ((a_|_ ^ b) v (a_|_ ^ b_|_))) = (a ->3 (b ->3 a))
29	24, 28	ax-r2 35	. . . 4 (((a_|_ ^ b) v (a_|_ ^ b_|_)) v a) = (a ->3 (b ->3 a))
30	23, 29	ax-r2 35	. . 3 (((a_|_ ^ (b ->1 a)) v (a_|_ ^ (b ->1 a)_|_)) v (a ^ (a_|_ v (b ->1 a)))) = (a ->3 (b ->3 a))
31	3, 30	ax-r2 35	. 2 (a ->3 (b ->1 a)) = (a ->3 (b ->3 a))
32	2, 31	ax-r2 35	1 (a ->3 ((b ->3 a_|_) ->3 b_|_)) = (a ->3 (b ->3 a))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->1 wi1 13   ->3 wi3 15

This theorem is referenced by:  u3lem14aa 774

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-i1 43  df-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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