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Theorem abid2f 2641
Description: A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.)
Hypothesis
Ref Expression
abid2f.1  |-  F/_ x A
Assertion
Ref Expression
abid2f  |-  { x  |  x  e.  A }  =  A

Proof of Theorem abid2f
StepHypRef Expression
1 nfab1 2615 . . 3  |-  F/_ x { x  |  x  e.  A }
2 abid2f.1 . . 3  |-  F/_ x A
31, 2cleqf 2639 . 2  |-  ( { x  |  x  e.  A }  =  A  <->  A. x ( x  e. 
{ x  |  x  e.  A }  <->  x  e.  A ) )
4 abid 2438 . 2  |-  ( x  e.  { x  |  x  e.  A }  <->  x  e.  A )
53, 4mpgbir 1596 1  |-  { x  |  x  e.  A }  =  A
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370    e. wcel 1758   {cab 2436   F/_wnfc 2599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601
This theorem is referenced by:  mptctf  26157  rabexgf  29886
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