Theorem rabid 2485

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 2315	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2	1	abeq2i 2148	1 |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) )

Syntax hints:   /\ wa 97   <-> wb 98   e. wcel 1393   {crab 2310

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-rab 2315

This theorem is referenced by:  rabeq2i  2554  rabn0m  3245  repizf2lem  3914  rabxfrd  4201  onintrab2im  4244  tfis  4306