Theorem rabeq2i 2683

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 2206	. 2 |- ( x e. A <-> x e. { x e. B | ph } )
3		rabid 2606	. 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 1331   e. wcel 1480   {crab 2420

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-rab 2425

This theorem is referenced by:  tfis  4497  fvmptssdm  5505  suplocsrlempr  7618  suplocsrlem  7619