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Theorem necon2ad 2312
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
StepHypRef Expression
1 necon2ad.1 . . 3  |-  ( ph  ->  ( A  =  B  ->  -.  ps )
)
21con2d 589 . 2  |-  ( ph  ->  ( ps  ->  -.  A  =  B )
)
3 df-ne 2256 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
42, 3syl6ibr 160 1  |-  ( ph  ->  ( ps  ->  A  =/=  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1289    =/= wne 2255
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 579  ax-in2 580
This theorem depends on definitions:  df-bi 115  df-ne 2256
This theorem is referenced by:  necon2d  2314  prneimg  3613  tz7.2  4172  nordeq  4350  pr2ne  6799  ltne  7549  apne  8076  xrltne  9247  ge0nemnf  9255  rpexp  11225  sqrt2irr  11234
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