Theorem necon2bd 2303

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 2246	. . 3 |- ( A =/= B <-> -. A = B )
3	1, 2	syl6ib 159	. 2 |- ( ph -> ( ps -> -. A = B ) )
4	3	con2d 586	1 |- ( ph -> ( A = B -> -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1284   =/= wne 2245

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-in1 576  ax-in2 577

This theorem depends on definitions:  df-bi 115  df-ne 2246

This theorem is referenced by:  nneo  8450  zeo2  8453  bezoutr1  10422  coprm  10523  sqrt2irr  10541