Theorem rabid 2670

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 2481	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2	1	abeq2i 2304	1 |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2164   {crab 2476

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-rab 2481

This theorem is referenced by:  rabeq2i  2757  rabn0m  3474  repizf2lem  4190  rabxfrd  4500  onintrab2im  4550  tfis  4615  nnwosdc  12176  imasnopn  14467