Metamath Proof Explorer


Theorem eliin2

Description: Membership in indexed intersection. See eliincex for a counterexample showing that the precondition B =/= (/) cannot be simply dropped. eliin uses an alternative precondition (and it doesn't have a disjoint var constraint between B and x ; see eliin2f ). (Contributed by Glauco Siliprandi, 26-Jun-2021)

Ref Expression
Assertion eliin2
|- ( B =/= (/) -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )

Proof

Step Hyp Ref Expression
1 nfcv
 |-  F/_ x B
2 1 eliin2f
 |-  ( B =/= (/) -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )