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Theorem eqneltrrd 2530
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
eqneltrrd.1  |-  ( ph  ->  A  =  B )
eqneltrrd.2  |-  ( ph  ->  -.  A  e.  C
)
Assertion
Ref Expression
eqneltrrd  |-  ( ph  ->  -.  B  e.  C
)

Proof of Theorem eqneltrrd
StepHypRef Expression
1 eqneltrrd.2 . 2  |-  ( ph  ->  -.  A  e.  C
)
2 eqneltrrd.1 . . 3  |-  ( ph  ->  A  =  B )
32eleq1d 2502 . 2  |-  ( ph  ->  ( A  e.  C  <->  B  e.  C ) )
41, 3mtbid 292 1  |-  ( ph  ->  -.  B  e.  C
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725
This theorem is referenced by:  bitsf1  12958  lssvancl2  16022  lbsind2  16153  lindfind2  27265  2atjlej  30276  2atnelvolN  30384
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-cleq 2429  df-clel 2432
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