Theorem i2i1 267

Description: Correspondence between Sasaki and Dishkant conditionals. (Contributed by NM, 7-Feb-1999.)

Assertion

Ref	Expression
i2i1	(a →2 b) = (b⊥ →1 a⊥ )

Proof of Theorem i2i1

Step	Hyp	Ref	Expression
1		ax-a1 30	. . 3 a = a⊥ ⊥
2	1	ud2lem0b 259	. 2 (a →2 b⊥ ⊥ ) = (a⊥ ⊥ →2 b⊥ ⊥ )
3		ax-a1 30	. . 3 b = b⊥ ⊥
4	3	ud2lem0a 258	. 2 (a →2 b) = (a →2 b⊥ ⊥ )
5		i1i2 266	. 2 (b⊥ →1 a⊥ ) = (a⊥ ⊥ →2 b⊥ ⊥ )
6	2, 4, 5	3tr1 63	1 (a →2 b) = (b⊥ →1 a⊥ )

Syntax hints:   = wb 1  ^⊥ wn 4   →₁ wi1 12   →₂ wi2 13

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-i1 44  df-i2 45

This theorem is referenced by:  nom40  325  nom41  326  nom42  327  nom43  328  nom44  329  nom45  330  nom50  331  nom51  332  nom52  333  nom53  334  nom54  335  nom55  336  oal2  999