Theorem abid2 2317

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 2311	. 2 |- A = { x | x e. A }
3	2	eqcomi 2200	1 |- { x | x e. A } = A

Syntax hints:   = wceq 1364   e. wcel 2167   {cab 2182

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192

This theorem is referenced by:  csbid  3092  abss  3252  ssab  3253  abssi  3258  notab  3433  inrab2  3436  dfrab2  3438  dfrab3  3439  notrab  3440  eusn  3696  dfopg  3806  iunid  3972  csbexga  4161  imai  5025  dffv4g  5555  frec0g  6455  dfixp  6759  euen1b  6862  acfun  7274  ccfunen  7331