Theorem abid2f 2345

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 2321	. . . . 5 |- F/_ x { x | x e. A }
3	1, 2	cleqf 2344	. . . 4 |- ( A = { x | x e. A } <-> A. x ( x e. A <-> x e. { x | x e. A } ) )
4		abid 2165	. . . . . 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 1470	. . . 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 1453	. 2 |- A = { x | x e. A }
10	9	eqcomi 2181	1 |- { x | x e. A } = A

Syntax hints:   <-> wb 105   A.wal 1351   = wceq 1353   e. wcel 2148   {cab 2163   F/_wnfc 2306

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308

This theorem is referenced by: (None)