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Theorem necon3abid 2294
Description: Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
Hypothesis
Ref Expression
necon3abid.1  |-  ( ph  ->  ( A  =  B  <->  ps ) )
Assertion
Ref Expression
necon3abid  |-  ( ph  ->  ( A  =/=  B  <->  -. 
ps ) )

Proof of Theorem necon3abid
StepHypRef Expression
1 df-ne 2256 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
2 necon3abid.1 . . 3  |-  ( ph  ->  ( A  =  B  <->  ps ) )
32notbid 627 . 2  |-  ( ph  ->  ( -.  A  =  B  <->  -.  ps )
)
41, 3syl5bb 190 1  |-  ( ph  ->  ( A  =/=  B  <->  -. 
ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    = wceq 1289    =/= wne 2255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580
This theorem depends on definitions:  df-bi 115  df-ne 2256
This theorem is referenced by:  necon3bbid  2295  fndmdif  5388  expnegap0  9927  gcdn0gt0  11049  cncongr2  11166
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