Theorem abid2 2275

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 170	. . 3 |- ( x e. A <-> x e. A )
2	1	abbi2i 2269	. 2 |- A = { x | x e. A }
3	2	eqcomi 2158	1 |- { x | x e. A } = A

Syntax hints:   = wceq 1332   e. wcel 2125   {cab 2140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-11 1483  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150

This theorem is referenced by:  csbid  3035  abss  3193  ssab  3194  abssi  3199  notab  3373  inrab2  3376  dfrab2  3378  dfrab3  3379  notrab  3380  eusn  3629  dfopg  3735  iunid  3900  csbexga  4088  imai  4935  dffv4g  5458  frec0g  6334  dfixp  6634  euen1b  6737  acfun  7121  ccfunen  7163