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Theorem abeq2d 2552
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 2510 . 2  |-  ( ph  ->  ( x  e.  A  <->  x  e.  { x  |  ps } ) )
3 abid 2431 . 2  |-  ( x  e.  { x  |  ps }  <->  ps )
42, 3syl6bb 254 1  |-  ( ph  ->  ( x  e.  A  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1654    e. wcel 1728   {cab 2429
This theorem is referenced by:  fvelimab  5818  ispridlc  26722  ac6s6  26800  dib1dim  32137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-11 1764  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439
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