Theorem abid2f 2374

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 2350	. . . . 5 |- F/_ x { x | x e. A }
3	1, 2	cleqf 2373	. . . 4 |- ( A = { x | x e. A } <-> A. x ( x e. A <-> x e. { x | x e. A } ) )
4		abid 2193	. . . . . 6 |- ( x e. { x | x e. A } <-> x e. A )
5	4	bibi2i 227	. . . . 5 |- ( ( x e. A <-> x e. { x | x e. A } ) <-> ( x e. A <-> x e. A ) )
6	5	albii 1493	. . . 4 |- ( A. x ( x e. A <-> x e. { x | x e. A } ) <-> A. x ( x e. A <-> x e. A ) )
7	3, 6	bitri 184	. . 3 |- ( A = { x | x e. A } <-> A. x ( x e. A <-> x e. A ) )
8		biid 171	. . 3 |- ( x e. A <-> x e. A )
9	7, 8	mpgbir 1476	. 2 |- A = { x | x e. A }
10	9	eqcomi 2209	1 |- { x | x e. A } = A

Syntax hints:   <-> wb 105   A.wal 1371   = wceq 1373   e. wcel 2176   {cab 2191   F/_wnfc 2335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337

This theorem is referenced by: (None)