Theorem abid2f 2260

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 2237	. . . . 5 |- F/_ x { x | x e. A }
3	1, 2	cleqf 2259	. . . 4 |- ( A = { x | x e. A } <-> A. x ( x e. A <-> x e. { x | x e. A } ) )
4		abid 2083	. . . . . 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 1411	. . . 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 1394	. 2 |- A = { x | x e. A }
10	9	eqcomi 2099	1 |- { x | x e. A } = A

Syntax hints:   <-> wb 104   A.wal 1294   = wceq 1296   e. wcel 1445   {cab 2081   F/_wnfc 2222

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224

This theorem is referenced by: (None)