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Theorem abeq2d 2539
Description: Equality of a class variable and a class abstraction (deduction form of abeq2 2537). (Contributed by NM, 16-Nov-1995.)
Hypothesis
Ref Expression
abeq2d.1  |-  ( ph  ->  A  =  { x  |  ps } )
Assertion
Ref Expression
abeq2d  |-  ( ph  ->  ( x  e.  A  <->  ps ) )

Proof of Theorem abeq2d
StepHypRef Expression
1 abeq2d.1 . . 3  |-  ( ph  ->  A  =  { x  |  ps } )
21eleq2d 2491 . 2  |-  ( ph  ->  ( x  e.  A  <->  x  e.  { x  |  ps } ) )
3 abid 2416 . 2  |-  ( x  e.  { x  |  ps }  <->  ps )
42, 3syl6bb 264 1  |-  ( ph  ->  ( x  e.  A  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1872   {cab 2414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-12 1909  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424
This theorem is referenced by:  abeq2i  2540  fvelimab  5881  ispridlc  32210  ac6s6  32322  dib1dim  34645
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