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Theorem abeq2d 2150
Description: Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)
Hypothesis
Ref Expression
abeqd.1  |-  ( ph  ->  A  =  { x  |  ps } )
Assertion
Ref Expression
abeq2d  |-  ( ph  ->  ( x  e.  A  <->  ps ) )

Proof of Theorem abeq2d
StepHypRef Expression
1 abeqd.1 . . 3  |-  ( ph  ->  A  =  { x  |  ps } )
21eleq2d 2107 . 2  |-  ( ph  ->  ( x  e.  A  <->  x  e.  { x  |  ps } ) )
3 abid 2028 . 2  |-  ( x  e.  { x  |  ps }  <->  ps )
42, 3syl6bb 185 1  |-  ( ph  ->  ( x  e.  A  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98    = wceq 1243    e. wcel 1393   {cab 2026
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-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036
This theorem is referenced by:  fvelimab  5229  frecsuclem3  5990
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