Theorem rabid 2580

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 2399	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2	1	abeq2i 2225	1 |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   e. wcel 1463   {crab 2394

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-rab 2399

This theorem is referenced by:  rabeq2i  2654  rabn0m  3356  repizf2lem  4045  rabxfrd  4350  onintrab2im  4394  tfis  4457  imasnopn  12310