Theorem i3th1 525

Description: Theorem for Kalmbach implication.

Assertion

Ref	Expression
i3th1	(a ->3 (a ->3 (b ->3 a))) = 1

Proof of Theorem i3th1

Step	Hyp	Ref	Expression
1		df2i3 480	. . 3 (b ->3 a) = ((b_|_ ^ a_|_) v ((b_|_ v a) ^ (b v (b_|_ ^ a))))
2	1	lor 66	. 2 (a_|_ v (b ->3 a)) = (a_|_ v ((b_|_ ^ a_|_) v ((b_|_ v a) ^ (b v (b_|_ ^ a)))))
3		lem4 493	. 2 (a ->3 (a ->3 (b ->3 a))) = (a_|_ v (b ->3 a))
4		ax-a3 31	. . . . 5 ((a_|_ v b) v (a ^ b_|_)) = (a_|_ v (b v (a ^ b_|_)))
5		anor1 80	. . . . . 6 (a ^ b_|_) = (a_|_ v b)_|_
6	5	lor 66	. . . . 5 ((a_|_ v b) v (a ^ b_|_)) = ((a_|_ v b) v (a_|_ v b)_|_)
7		ax-a3 31	. . . . . . . 8 ((a_|_ v (a_|_ ^ b)) v ((b_|_ v a) ^ (b v (b_|_ ^ a)))) = (a_|_ v ((a_|_ ^ b) v ((b_|_ v a) ^ (b v (b_|_ ^ a)))))
8		ax-a2 30	. . . . . . . . . . . . 13 (b_|_ v a) = (a v b_|_)
9		anor2 81	. . . . . . . . . . . . . . 15 (a_|_ ^ b) = (a v b_|_)_|_
10	9	con2 64	. . . . . . . . . . . . . 14 (a_|_ ^ b)_|_ = (a v b_|_)
11	10	ax-r1 34	. . . . . . . . . . . . 13 (a v b_|_) = (a_|_ ^ b)_|_
12	8, 11	ax-r2 35	. . . . . . . . . . . 12 (b_|_ v a) = (a_|_ ^ b)_|_
13		ancom 68	. . . . . . . . . . . . 13 (b_|_ ^ a) = (a ^ b_|_)
14	13	lor 66	. . . . . . . . . . . 12 (b v (b_|_ ^ a)) = (b v (a ^ b_|_))
15	12, 14	2an 72	. . . . . . . . . . 11 ((b_|_ v a) ^ (b v (b_|_ ^ a))) = ((a_|_ ^ b)_|_ ^ (b v (a ^ b_|_)))
16	15	lor 66	. . . . . . . . . 10 ((a_|_ ^ b) v ((b_|_ v a) ^ (b v (b_|_ ^ a)))) = ((a_|_ ^ b) v ((a_|_ ^ b)_|_ ^ (b v (a ^ b_|_))))
17		oml5 431	. . . . . . . . . 10 ((a_|_ ^ b) v ((a_|_ ^ b)_|_ ^ (b v (a ^ b_|_)))) = (b v (a ^ b_|_))
18	16, 17	ax-r2 35	. . . . . . . . 9 ((a_|_ ^ b) v ((b_|_ v a) ^ (b v (b_|_ ^ a)))) = (b v (a ^ b_|_))
19	18	lor 66	. . . . . . . 8 (a_|_ v ((a_|_ ^ b) v ((b_|_ v a) ^ (b v (b_|_ ^ a))))) = (a_|_ v (b v (a ^ b_|_)))
20	7, 19	ax-r2 35	. . . . . . 7 ((a_|_ v (a_|_ ^ b)) v ((b_|_ v a) ^ (b v (b_|_ ^ a)))) = (a_|_ v (b v (a ^ b_|_)))
21	20	ax-r1 34	. . . . . 6 (a_|_ v (b v (a ^ b_|_))) = ((a_|_ v (a_|_ ^ b)) v ((b_|_ v a) ^ (b v (b_|_ ^ a))))
22		a5b 112	. . . . . . 7 (a_|_ v (a_|_ ^ b)) = a_|_
23	22	ax-r5 37	. . . . . 6 ((a_|_ v (a_|_ ^ b)) v ((b_|_ v a) ^ (b v (b_|_ ^ a)))) = (a_|_ v ((b_|_ v a) ^ (b v (b_|_ ^ a))))
24	21, 23	ax-r2 35	. . . . 5 (a_|_ v (b v (a ^ b_|_))) = (a_|_ v ((b_|_ v a) ^ (b v (b_|_ ^ a))))
25	4, 6, 24	3tr2 61	. . . 4 ((a_|_ v b) v (a_|_ v b)_|_) = (a_|_ v ((b_|_ v a) ^ (b v (b_|_ ^ a))))
26		df-t 40	. . . 4 1 = ((a_|_ v b) v (a_|_ v b)_|_)
27		ancom 68	. . . . . . 7 (b_|_ ^ a_|_) = (a_|_ ^ b_|_)
28	27	lor 66	. . . . . 6 (a_|_ v (b_|_ ^ a_|_)) = (a_|_ v (a_|_ ^ b_|_))
29		a5b 112	. . . . . 6 (a_|_ v (a_|_ ^ b_|_)) = a_|_
30	28, 29	ax-r2 35	. . . . 5 (a_|_ v (b_|_ ^ a_|_)) = a_|_
31	30	ax-r5 37	. . . 4 ((a_|_ v (b_|_ ^ a_|_)) v ((b_|_ v a) ^ (b v (b_|_ ^ a)))) = (a_|_ v ((b_|_ v a) ^ (b v (b_|_ ^ a))))
32	25, 26, 31	3tr1 60	. . 3 1 = ((a_|_ v (b_|_ ^ a_|_)) v ((b_|_ v a) ^ (b v (b_|_ ^ a))))
33		ax-a3 31	. . 3 ((a_|_ v (b_|_ ^ a_|_)) v ((b_|_ v a) ^ (b v (b_|_ ^ a)))) = (a_|_ v ((b_|_ ^ a_|_) v ((b_|_ v a) ^ (b v (b_|_ ^ a)))))
34	32, 33	ax-r2 35	. 2 1 = (a_|_ v ((b_|_ ^ a_|_) v ((b_|_ v a) ^ (b v (b_|_ ^ a)))))
35	2, 3, 34	3tr1 60	1 (a ->3 (a ->3 (b ->3 a))) = 1

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->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-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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