Theorem abid2f 2307

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 2284	. . . . 5 |- F/_ x { x | x e. A }
3	1, 2	cleqf 2306	. . . 4 |- ( A = { x | x e. A } <-> A. x ( x e. A <-> x e. { x | x e. A } ) )
4		abid 2128	. . . . . 6 |- ( x e. { x | x e. A } <-> x e. A )
5	4	bibi2i 226	. . . . 5 |- ( ( x e. A <-> x e. { x | x e. A } ) <-> ( x e. A <-> x e. A ) )
6	5	albii 1447	. . . 4 |- ( A. x ( x e. A <-> x e. { x | x e. A } ) <-> A. x ( x e. A <-> x e. A ) )
7	3, 6	bitri 183	. . 3 |- ( A = { x | x e. A } <-> A. x ( x e. A <-> x e. A ) )
8		biid 170	. . 3 |- ( x e. A <-> x e. A )
9	7, 8	mpgbir 1430	. 2 |- A = { x | x e. A }
10	9	eqcomi 2144	1 |- { x | x e. A } = A

Syntax hints:   <-> wb 104   A.wal 1330   = wceq 1332   e. wcel 1481   {cab 2126   F/_wnfc 2269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271

This theorem is referenced by: (None)