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Theorem eqneltrd 2133
Description: If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
eqneltrd.1  |-  ( ph  ->  A  =  B )
eqneltrd.2  |-  ( ph  ->  -.  B  e.  C
)
Assertion
Ref Expression
eqneltrd  |-  ( ph  ->  -.  A  e.  C
)

Proof of Theorem eqneltrd
StepHypRef Expression
1 eqneltrd.2 . 2  |-  ( ph  ->  -.  B  e.  C
)
2 eqneltrd.1 . . 3  |-  ( ph  ->  A  =  B )
32eleq1d 2106 . 2  |-  ( ph  ->  ( A  e.  C  <->  B  e.  C ) )
41, 3mtbird 598 1  |-  ( ph  ->  -.  A  e.  C
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1243    e. wcel 1393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036
This theorem is referenced by: (None)
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