Theorem abid2 2353

Description: A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)

Assertion

Ref	Expression
abid2	|- { x | x e. A } = A

Distinct variable group:   x, A

Proof of Theorem abid2

Step	Hyp	Ref	Expression
1		biid 171	. . 3 |- ( x e. A <-> x e. A )
2	1	abbi2i 2346	. 2 |- A = { x | x e. A }
3	2	eqcomi 2235	1 |- { x | x e. A } = A

Syntax hints:   = wceq 1398   e. wcel 2202   {cab 2217

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-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227

This theorem is referenced by:  csbid  3136  abss  3297  ssab  3298  abssi  3303  notab  3479  inrab2  3482  dfrab2  3484  dfrab3  3485  notrab  3486  eusn  3749  dfopg  3865  iunid  4031  csbexga  4222  imai  5099  dffv4g  5645  frec0g  6606  dfixp  6912  euen1b  7020  modom2  7038  acfun  7465  ccfunen  7526