Theorem i4i3 263

Description: Correspondence between Kalmbach and non-tollens conditionals.

Assertion

Ref	Expression
i4i3	(a →4 b) = (b⊥ →3 a⊥ )

Proof of Theorem i4i3

Step	Hyp	Ref	Expression
1		ax-a1 29	. . . 4 b = b⊥ ⊥
2	1	ud4lem0a 254	. . 3 (a →4 b) = (a →4 b⊥ ⊥ )
3		ax-a1 29	. . . 4 a = a⊥ ⊥
4	3	ud4lem0b 255	. . 3 (a →4 b⊥ ⊥ ) = (a⊥ ⊥ →4 b⊥ ⊥ )
5	2, 4	ax-r2 35	. 2 (a →4 b) = (a⊥ ⊥ →4 b⊥ ⊥ )
6		i3i4 262	. . 3 (b⊥ →3 a⊥ ) = (a⊥ ⊥ →4 b⊥ ⊥ )
7	6	ax-r1 34	. 2 (a⊥ ⊥ →4 b⊥ ⊥ ) = (b⊥ →3 a⊥ )
8	5, 7	ax-r2 35	1 (a →4 b) = (b⊥ →3 a⊥ )

Syntax hints:   = wb 1  ^⊥ wn 4   →₃ wi3 15   →₄ wi4 16

This theorem is referenced by:  nom44 321  dfi4b 482

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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