Theorem abid2f 2791

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

Step	Hyp	Ref	Expression
1		nfab1 2766	. . 3 |- F/_ x { x | x e. A }
2		abid2f.1	. . 3 |- F/_ x A
3	1, 2	cleqf 2790	. 2 |- ( { x | x e. A } = A <-> A. x ( x e. { x | x e. A } <-> x e. A ) )
4		abid 2610	. 2 |- ( x e. { x | x e. A } <-> x e. A )
5	3, 4	mpgbir 1726	1 |- { x | x e. A } = A

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   {cab 2608   F/_wnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by:  mptctf  29495  rabexgf  39183  ssabf  39280  abssf  39295