Theorem i3i4 262

Description: Correspondence between Kalmbach and non-tollens conditionals.

Assertion

Ref	Expression
i3i4	(a ->3 b) = (b_|_ ->4 a_|_)

Proof of Theorem i3i4

Step	Hyp	Ref	Expression
1		ax-a2 30	. . . 4 ((a_|_ ^ b) v (a_|_ ^ b_|_)) = ((a_|_ ^ b_|_) v (a_|_ ^ b))
2		ancom 68	. . . . 5 (a_|_ ^ b_|_) = (b_|_ ^ a_|_)
3		ancom 68	. . . . . 6 (a_|_ ^ b) = (b ^ a_|_)
4		ax-a1 29	. . . . . . 7 b = b_|__|_
5	4	ran 71	. . . . . 6 (b ^ a_|_) = (b_|__|_ ^ a_|_)
6	3, 5	ax-r2 35	. . . . 5 (a_|_ ^ b) = (b_|__|_ ^ a_|_)
7	2, 6	2or 67	. . . 4 ((a_|_ ^ b_|_) v (a_|_ ^ b)) = ((b_|_ ^ a_|_) v (b_|__|_ ^ a_|_))
8	1, 7	ax-r2 35	. . 3 ((a_|_ ^ b) v (a_|_ ^ b_|_)) = ((b_|_ ^ a_|_) v (b_|__|_ ^ a_|_))
9		ancom 68	. . . 4 (a ^ (a_|_ v b)) = ((a_|_ v b) ^ a)
10		ax-a2 30	. . . . . 6 (a_|_ v b) = (b v a_|_)
11	4	ax-r5 37	. . . . . 6 (b v a_|_) = (b_|__|_ v a_|_)
12	10, 11	ax-r2 35	. . . . 5 (a_|_ v b) = (b_|__|_ v a_|_)
13		ax-a1 29	. . . . 5 a = a_|__|_
14	12, 13	2an 72	. . . 4 ((a_|_ v b) ^ a) = ((b_|__|_ v a_|_) ^ a_|__|_)
15	9, 14	ax-r2 35	. . 3 (a ^ (a_|_ v b)) = ((b_|__|_ v a_|_) ^ a_|__|_)
16	8, 15	2or 67	. 2 (((a_|_ ^ b) v (a_|_ ^ b_|_)) v (a ^ (a_|_ v b))) = (((b_|_ ^ a_|_) v (b_|__|_ ^ a_|_)) v ((b_|__|_ v a_|_) ^ a_|__|_))
17		df-i3 45	. 2 (a ->3 b) = (((a_|_ ^ b) v (a_|_ ^ b_|_)) v (a ^ (a_|_ v b)))
18		df-i4 46	. 2 (b_|_ ->4 a_|_) = (((b_|_ ^ a_|_) v (b_|__|_ ^ a_|_)) v ((b_|__|_ v a_|_) ^ a_|__|_))
19	16, 17, 18	3tr1 60	1 (a ->3 b) = (b_|_ ->4 a_|_)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->3 wi3 15   ->4 wi4 16

This theorem is referenced by:  i4i3 263  nom43 320

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-i3 45  df-i4 46

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