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Theorem eqneltrd 2409
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 2382 . 2  |-  ( ph  ->  ( A  e.  C  <->  B  e.  C ) )
41, 3mtbird 292 1  |-  ( ph  ->  -.  A  e.  C
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1633    e. wcel 1701
This theorem is referenced by:  mreexmrid  13594  dvreasin  25340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-11 1732  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1533  df-cleq 2309  df-clel 2312
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