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Theorem neleqtrd 2185
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 2157 . 2  |-  ( ph  ->  ( C  e.  A  <->  C  e.  B ) )
41, 3mtbid 632 1  |-  ( ph  ->  -.  C  e.  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1289    e. wcel 1438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-clel 2084
This theorem is referenced by: (None)
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