Theorem rabeq2i 2554

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 2104	. 2 |- ( x e. A <-> x e. { x e. B | ph } )
3		rabid 2485	. 2 |- ( x e. { x e. B | ph } <-> ( x e. B /\ ph ) )
4	2, 3	bitri 173	1 |- ( x e. A <-> ( x e. B /\ ph ) )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   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:  tfis  4306  fvmptssdm  5255