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	$⊢ φ \to J \in Top$
		opnneilv.2	$⊢ (φ \land y \subseteq x) \to (ψ \to χ)$
		opnneil.3	$⊢ (φ \land x = y) \to (ψ \leftrightarrow χ)$
	Assertion	opnneieqv	$⊢ φ \to (\exists x \in nei (J) (S) ψ \leftrightarrow \exists x \in J (S \subseteq x \land ψ))$

Proof

Step	Hyp	Ref	Expression
1		opnneir.1	$⊢ φ \to J \in Top$
2		opnneilv.2	$⊢ (φ \land y \subseteq x) \to (ψ \to χ)$
3		opnneil.3	$⊢ (φ \land x = y) \to (ψ \leftrightarrow χ)$
4	1 2 3	opnneil	$⊢ φ \to (\exists x \in nei (J) (S) ψ \to \exists x \in J (S \subseteq x \land ψ))$
5	1	opnneir	$⊢ φ \to (\exists x \in J (S \subseteq x \land ψ) \to \exists x \in nei (J) (S) ψ)$
6	4 5	impbid	$⊢ φ \to (\exists x \in nei (J) (S) ψ \leftrightarrow \exists x \in J (S \subseteq x \land ψ))$