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Theorem abid2f 2645
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 2618 . . 3  |-  F/_ x { x  |  x  e.  A }
2 abid2f.1 . . 3  |-  F/_ x A
31, 2cleqf 2643 . 2  |-  ( { x  |  x  e.  A }  =  A  <->  A. x ( x  e. 
{ x  |  x  e.  A }  <->  x  e.  A ) )
4 abid 2441 . 2  |-  ( x  e.  { x  |  x  e.  A }  <->  x  e.  A )
53, 4mpgbir 1627 1  |-  { x  |  x  e.  A }  =  A
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1398    e. wcel 1823   {cab 2439   F/_wnfc 2602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604
This theorem is referenced by:  mptctf  27774  rabexgf  31639
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