Theorem necon2ad 2302

Description: Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)

Hypothesis

Ref	Expression
necon2ad.1	|- ( ph -> ( A = B -> -. ps ) )

Assertion

Ref	Expression
necon2ad	|- ( ph -> ( ps -> A =/= B ) )

Proof of Theorem necon2ad

Step	Hyp	Ref	Expression
1		necon2ad.1	. . 3 |- ( ph -> ( A = B -> -. ps ) )
2	1	con2d 586	. 2 |- ( ph -> ( ps -> -. A = B ) )
3		df-ne 2246	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	syl6ibr 160	1 |- ( ph -> ( ps -> A =/= B ) )

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-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577

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

This theorem is referenced by:  necon2d  2304  prneimg  3566  tz7.2  4109  nordeq  4287  pr2ne  6461  ltne  7196  apne  7723  xrltne  8883  ge0nemnf  8891  rpexp  10532  sqrt2irr  10541