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Theorem necon2bd 2398
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 2341 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
31, 2syl6ib 160 . 2  |-  ( ph  ->  ( ps  ->  -.  A  =  B )
)
43con2d 619 1  |-  ( ph  ->  ( A  =  B  ->  -.  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1348    =/= wne 2340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 609  ax-in2 610
This theorem depends on definitions:  df-bi 116  df-ne 2341
This theorem is referenced by:  disjiun  3982  map0g  6662  nneo  9302  zeo2  9305  bezoutr1  11975  coprm  12085  sqrt2irr  12103  dfphi2  12161  bj-charfunr  13805  nconstwlpolem  14056
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