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Theorem necon1abid 2499
Description: Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.)
Hypothesis
Ref Expression
necon1abid.1  |-  ( ph  ->  ( -.  ps  <->  A  =  B ) )
Assertion
Ref Expression
necon1abid  |-  ( ph  ->  ( A  =/=  B  <->  ps ) )

Proof of Theorem necon1abid
StepHypRef Expression
1 df-ne 2448 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
2 necon1abid.1 . . 3  |-  ( ph  ->  ( -.  ps  <->  A  =  B ) )
32con1bid 320 . 2  |-  ( ph  ->  ( -.  A  =  B  <->  ps ) )
41, 3syl5bb 248 1  |-  ( ph  ->  ( A  =/=  B  <->  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    = wceq 1623    =/= wne 2446
This theorem is referenced by:  lttri2  8904  xrlttri2  10476  ioon0  10682  lssne0  15708  xmetgt0  17922
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-ne 2448
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