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Theorem nemtbir 2547
Description: An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
Hypotheses
Ref Expression
nemtbir.1  |-  A  =/= 
B
nemtbir.2  |-  ( ph  <->  A  =  B )
Assertion
Ref Expression
nemtbir  |-  -.  ph

Proof of Theorem nemtbir
StepHypRef Expression
1 nemtbir.1 . . 3  |-  A  =/= 
B
2 df-ne 2461 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
31, 2mpbi 199 . 2  |-  -.  A  =  B
4 nemtbir.2 . 2  |-  ( ph  <->  A  =  B )
53, 4mtbir 290 1  |-  -.  ph
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1632    =/= wne 2459
This theorem is referenced by:  opthwiener  4284  opthprc  4752  cfpwsdom  8222  gzrngunitlem  16452  ex-id  20837  sltval2  24381  sltsolem1  24393
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 2461
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