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Theorem eqnetrd 2307
Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
Hypotheses
Ref Expression
eqnetrd.1  |-  ( ph  ->  A  =  B )
eqnetrd.2  |-  ( ph  ->  B  =/=  C )
Assertion
Ref Expression
eqnetrd  |-  ( ph  ->  A  =/=  C )

Proof of Theorem eqnetrd
StepHypRef Expression
1 eqnetrd.2 . 2  |-  ( ph  ->  B  =/=  C )
2 eqnetrd.1 . . 3  |-  ( ph  ->  A  =  B )
32neeq1d 2301 . 2  |-  ( ph  ->  ( A  =/=  C  <->  B  =/=  C ) )
41, 3mpbird 166 1  |-  ( ph  ->  A  =/=  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    =/= wne 2283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-5 1406  ax-gen 1408  ax-4 1470  ax-17 1489  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-cleq 2108  df-ne 2284
This theorem is referenced by:  eqnetrrd  2309  frecabcl  6262  frecsuclem  6269  omp1eomlem  6945  xaddnemnf  9580  xaddnepnf  9581  hashprg  10494  bezoutr1  11617  phibndlem  11787  dfphi2  11791
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