Theorem rabeq2i 2757

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 2260	. 2 |- ( x e. A <-> x e. { x e. B | ph } )
3		rabid 2670	. 2 |- ( x e. { x e. B | ph } <-> ( x e. B /\ ph ) )
4	2, 3	bitri 184	1 |- ( x e. A <-> ( x e. B /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   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:  tfis  4611  fvmptssdm  5634  suplocsrlempr  7857  suplocsrlem  7858