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Theorem abeq2d 2497
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 2455 . 2  |-  ( ph  ->  ( x  e.  A  <->  x  e.  { x  |  ps } ) )
3 abid 2376 . 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 1649    e. wcel 1717   {cab 2374
This theorem is referenced by:  fvelimab  5722  ispridlc  26372  dib1dim  31281
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 2369
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384
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