Metamath Proof Explorer


Theorem opnneieqv

Description: The equivalence between neighborhood and open neighborhood. See opnneieqvv for different dummy variables. (Contributed by Zhi Wang, 31-Aug-2024)

Ref Expression
Hypotheses opnneir.1 φ J Top
opnneilv.2 φ y x ψ χ
opnneil.3 φ x = y ψ χ
Assertion opnneieqv φ x nei J S ψ x J S x ψ

Proof

Step Hyp Ref Expression
1 opnneir.1 φ J Top
2 opnneilv.2 φ y x ψ χ
3 opnneil.3 φ x = y ψ χ
4 1 2 3 opnneil φ x nei J S ψ x J S x ψ
5 1 opnneir φ x J S x ψ x nei J S ψ
6 4 5 impbid φ x nei J S ψ x J S x ψ