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  3066  abss  3225  ssab  3226  abssi  3231  notab  3406  inrab2  3409  dfrab2  3411  dfrab3  3412  notrab  3413  eusn  3667  dfopg  3777  iunid  3943  csbexga  4132  imai  4985  dffv4g  5513  frec0g  6398  dfixp  6700  euen1b  6803  acfun  7206  ccfunen  7263