Theorem abid2 2298

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 2292	. 2 |- A = { x | x e. A }
3	2	eqcomi 2181	1 |- { x | x e. A } = A

Syntax hints:   = wceq 1353   e. wcel 2148   {cab 2163

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  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-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173

This theorem is referenced by:  csbid  3065  abss  3224  ssab  3225  abssi  3230  notab  3405  inrab2  3408  dfrab2  3410  dfrab3  3411  notrab  3412  eusn  3666  dfopg  3776  iunid  3942  csbexga  4131  imai  4984  dffv4g  5512  frec0g  6397  dfixp  6699  euen1b  6802  acfun  7205  ccfunen  7262