Theorem rabid 2529

Description: An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.)

Assertion

Ref	Expression
rabid	|- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) )

Proof of Theorem rabid

Step	Hyp	Ref	Expression
1		df-rab 2357	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2	1	abeq2i 2189	1 |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) )

Syntax hints:   /\ wa 102   <-> wb 103   e. wcel 1433   {crab 2352

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-rab 2357

This theorem is referenced by:  rabeq2i  2598  rabn0m  3272  repizf2lem  3935  rabxfrd  4219  onintrab2im  4262  tfis  4324