Theorem rabeq2i 2727

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

Hypothesis

Ref	Expression
rabeq2i.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		rabeq2i.1	. . 3 |- A = { x e. B | ph }
2	1	eleq2i 2237	. 2 |- ( x e. A <-> x e. { x e. B | ph } )
3		rabid 2645	. 2 |- ( x e. { x e. B | ph } <-> ( x e. B /\ ph ) )
4	2, 3	bitri 183	1 |- ( x e. A <-> ( x e. B /\ ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   {crab 2452

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-rab 2457

This theorem is referenced by:  tfis  4567  fvmptssdm  5580  suplocsrlempr  7769  suplocsrlem  7770