Theorem i4i3 271

Description: Correspondence between Kalmbach and non-tollens conditionals. (Contributed by NM, 7-Feb-1999.)

Assertion

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

Proof of Theorem i4i3

Step	Hyp	Ref	Expression
1		ax-a1 30	. . . 4 b = b⊥ ⊥
2	1	ud4lem0a 262	. . 3 (a →4 b) = (a →4 b⊥ ⊥ )
3		ax-a1 30	. . . 4 a = a⊥ ⊥
4	3	ud4lem0b 263	. . 3 (a →4 b⊥ ⊥ ) = (a⊥ ⊥ →4 b⊥ ⊥ )
5	2, 4	ax-r2 36	. 2 (a →4 b) = (a⊥ ⊥ →4 b⊥ ⊥ )
6		i3i4 270	. . 3 (b⊥ →3 a⊥ ) = (a⊥ ⊥ →4 b⊥ ⊥ )
7	6	ax-r1 35	. 2 (a⊥ ⊥ →4 b⊥ ⊥ ) = (b⊥ →3 a⊥ )
8	5, 7	ax-r2 36	1 (a →4 b) = (b⊥ →3 a⊥ )

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   →₃ wi3 14   →₄ wi4 15

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This proof depends on definitions:  df-a 40  df-i3 46  df-i4 47

This theorem is used by:  nom44  329  dfi4b  500