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Theorem necon2bd 2313
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
StepHypRef Expression
1 necon2bd.1 . . 3  |-  ( ph  ->  ( ps  ->  A  =/=  B ) )
2 df-ne 2256 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
31, 2syl6ib 159 . 2  |-  ( ph  ->  ( ps  ->  -.  A  =  B )
)
43con2d 589 1  |-  ( ph  ->  ( A  =  B  ->  -.  ps )
)
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-in1 579  ax-in2 580
This theorem depends on definitions:  df-bi 115  df-ne 2256
This theorem is referenced by:  disjiun  3832  map0g  6425  nneo  8819  zeo2  8822  bezoutr1  11104  coprm  11205  sqrt2irr  11223  dfphi2  11278
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