Theorem rabid 2641

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 2453	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2	1	abeq2i 2277	1 |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   e. wcel 2136   {crab 2448

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-rab 2453

This theorem is referenced by:  rabeq2i  2723  rabn0m  3436  repizf2lem  4140  rabxfrd  4447  onintrab2im  4495  tfis  4560  nnwosdc  11972  imasnopn  12939