Metamath Proof Explorer


Theorem rabeq2i

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

Ref Expression
Hypothesis rabeq2i.1
|- A = { x e. B | ph }
Assertion rabeq2i
|- ( x e. A <-> ( x e. B /\ ph ) )

Proof

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