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	|- ( ph -> J e. Top )
	opnneilv.2	|- ( ( ph /\ y C_ x ) -> ( ps -> ch ) )
	opnneil.3	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
Assertion	opnneieqv	|- ( ph -> ( E. x e. ( ( nei ` J ) ` S ) ps <-> E. x e. J ( S C_ x /\ ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		opnneir.1	|- ( ph -> J e. Top )
2		opnneilv.2	|- ( ( ph /\ y C_ x ) -> ( ps -> ch ) )
3		opnneil.3	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
4	1 2 3	opnneil	|- ( ph -> ( E. x e. ( ( nei ` J ) ` S ) ps -> E. x e. J ( S C_ x /\ ps ) ) )
5	1	opnneir	|- ( ph -> ( E. x e. J ( S C_ x /\ ps ) -> E. x e. ( ( nei ` J ) ` S ) ps ) )
6	4 5	impbid	|- ( ph -> ( E. x e. ( ( nei ` J ) ` S ) ps <-> E. x e. J ( S C_ x /\ ps ) ) )