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Theorem neleqtrd 2482
Description: If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
neleqtrd.1  |-  ( ph  ->  -.  C  e.  A
)
neleqtrd.2  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
neleqtrd  |-  ( ph  ->  -.  C  e.  B
)

Proof of Theorem neleqtrd
StepHypRef Expression
1 neleqtrd.1 . 2  |-  ( ph  ->  -.  C  e.  A
)
2 neleqtrd.2 . . 3  |-  ( ph  ->  A  =  B )
32eleq2d 2454 . 2  |-  ( ph  ->  ( C  e.  A  <->  C  e.  B ) )
41, 3mtbid 292 1  |-  ( ph  ->  -.  C  e.  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1717
This theorem is referenced by:  smoord  6563  r1tskina  8590  mreexexlem2d  13797  stoweidlem26  27443  dochnel  31508
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-11 1753  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-cleq 2380  df-clel 2383
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