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Theorem abeq2d 2544
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 2502 . 2  |-  ( ph  ->  ( x  e.  A  <->  x  e.  { x  |  ps } ) )
3 abid 2423 . 2  |-  ( x  e.  { x  |  ps }  <->  ps )
42, 3syl6bb 253 1  |-  ( ph  ->  ( x  e.  A  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   {cab 2421
This theorem is referenced by:  fvelimab  5774  ispridlc  26634  dib1dim  31864
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 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431
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