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Theorem neeq1d 2273
Description: Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
Hypothesis
Ref Expression
neeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
neeq1d  |-  ( ph  ->  ( A  =/=  C  <->  B  =/=  C ) )

Proof of Theorem neeq1d
StepHypRef Expression
1 neeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 neeq1 2268 . 2  |-  ( A  =  B  ->  ( A  =/=  C  <->  B  =/=  C ) )
31, 2syl 14 1  |-  ( ph  ->  ( A  =/=  C  <->  B  =/=  C ) )
Colors of variables: wff set class
Syntax hints:    -> 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  ax-5 1381  ax-gen 1383  ax-4 1445  ax-17 1464  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-ne 2256
This theorem is referenced by:  neeq12d  2275  eqnetrd  2279  prnzg  3562  hashprg  10204  ialgcvg  11295  ialgcvga  11298  eucialgcvga  11305  rpdvds  11346  phibndlem  11457  dfphi2  11461
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