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Theorem necon1bbid 2606
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 2554 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
2 necon1bbid.1 . . 3  |-  ( ph  ->  ( A  =/=  B  <->  ps ) )
31, 2syl5bbr 251 . 2  |-  ( ph  ->  ( -.  A  =  B  <->  ps ) )
43con1bid 321 1  |-  ( ph  ->  ( -.  ps  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    = wceq 1649    =/= wne 2552
This theorem is referenced by:  blssioo  18699  metdstri  18754  dchrpt  20920  lgsquad3  21014  eupath2lem2  21550  lkrpssN  29280  dochshpsat  31571
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 178  df-ne 2554
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