Theorem necon2bd 2566

Description: Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)

Hypothesis

Ref	Expression
necon2bd.1	|- (ph -> (ps -> A =/= B))

Assertion

Ref	Expression
necon2bd	|- (ph -> (A = B -> -. ps))

Proof of Theorem necon2bd

Step	Hyp	Ref	Expression
1		necon2bd.1	. . 3 |- (ph -> (ps -> A =/= B))
2		df-ne 2519	. . 3 |- (A =/= B <-> -. A = B)
3	1, 2	syl6ib 217	. 2 |- (ph -> (ps -> -. A = B))
4	3	con2d 107	1 |- (ph -> (A = B -> -. ps))

Syntax hints:  -. wn 3   -> wi 4   = wceq 1642   =/= wne 2517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-ne 2519

This theorem is referenced by:  necon4d  2580