Theorem abid2f 2514

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/_xA

Assertion

Ref	Expression
abid2f	|- {x | x e. A} = A

Proof of Theorem abid2f

Step	Hyp	Ref	Expression
1		abid2f.1	. . . . 5 |- F/_xA
2		nfab1 2491	. . . . 5 |- F/_x{x | x e. A}
3	1, 2	cleqf 2513	. . . 4 |- (A = {x | x e. A} <-> A.x(x e. A <-> x e. {x | x e. A}))
4		abid 2341	. . . . . 6 |- (x e. {x | x e. A} <-> x e. A)
5	4	bibi2i 304	. . . . 5 |- ((x e. A <-> x e. {x | x e. A}) <-> (x e. A <-> x e. A))
6	5	albii 1566	. . . 4 |- (A.x(x e. A <-> x e. {x | x e. A}) <-> A.x(x e. A <-> x e. A))
7	3, 6	bitri 240	. . 3 |- (A = {x | x e. A} <-> A.x(x e. A <-> x e. A))
8		biid 227	. . 3 |- (x e. A <-> x e. A)
9	7, 8	mpgbir 1550	. 2 |- A = {x | x e. A}
10	9	eqcomi 2357	1 |- {x | x e. A} = A

Syntax hints:   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710  {cab 2339   F/_wnfc 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478

This theorem is referenced by: (None)