Theorem rabeq2i 2598

Description: Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)

Hypothesis

Ref	Expression
rabeqi.1	|- A = { x e. B | ph }

Assertion

Ref	Expression
rabeq2i	|- ( x e. A <-> ( x e. B /\ ph ) )

Proof of Theorem rabeq2i

Step	Hyp	Ref	Expression
1		rabeqi.1	. . 3 |- A = { x e. B | ph }
2	1	eleq2i 2145	. 2 |- ( x e. A <-> x e. { x e. B | ph } )
3		rabid 2529	. 2 |- ( x e. { x e. B | ph } <-> ( x e. B /\ ph ) )
4	2, 3	bitri 182	1 |- ( x e. A <-> ( x e. B /\ ph ) )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   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:  tfis  4324  fvmptssdm  5276