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Theorem necon1bbid 2500
Description: Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.)
Hypothesis
Ref Expression
necon1bbid.1  |-  ( ph  ->  ( A  =/=  B  <->  ps ) )
Assertion
Ref Expression
necon1bbid  |-  ( ph  ->  ( -.  ps  <->  A  =  B ) )

Proof of Theorem necon1bbid
StepHypRef Expression
1 df-ne 2448 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
2 necon1bbid.1 . . 3  |-  ( ph  ->  ( A  =/=  B  <->  ps ) )
31, 2syl5bbr 250 . 2  |-  ( ph  ->  ( -.  A  =  B  <->  ps ) )
43con1bid 320 1  |-  ( ph  ->  ( -.  ps  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    = wceq 1623    =/= wne 2446
This theorem is referenced by:  blssioo  18301  metdstri  18355  dchrpt  20506  lgsquad3  20600  eupath2lem2  23902  lkrpssN  29353  dochshpsat  31644
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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