Theorem abid2f 2243

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.)

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		abid2f.1	. . . . 5 |- F/_ x A
2		nfab1 2221	. . . . 5 |- F/_ x { x | x e. A }
3	1, 2	cleqf 2242	. . . 4 |- ( A = { x | x e. A } <-> A. x ( x e. A <-> x e. { x | x e. A } ) )
4		abid 2069	. . . . . 6 |- ( x e. { x | x e. A } <-> x e. A )
5	4	bibi2i 225	. . . . 5 |- ( ( x e. A <-> x e. { x | x e. A } ) <-> ( x e. A <-> x e. A ) )
6	5	albii 1399	. . . 4 |- ( A. x ( x e. A <-> x e. { x | x e. A } ) <-> A. x ( x e. A <-> x e. A ) )
7	3, 6	bitri 182	. . 3 |- ( A = { x | x e. A } <-> A. x ( x e. A <-> x e. A ) )
8		biid 169	. . 3 |- ( x e. A <-> x e. A )
9	7, 8	mpgbir 1382	. 2 |- A = { x | x e. A }
10	9	eqcomi 2085	1 |- { x | x e. A } = A

Syntax hints:   <-> wb 103   A.wal 1282   = wceq 1284   e. wcel 1433   {cab 2067   F/_wnfc 2206

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208

This theorem is referenced by: (None)