Theorem opnneirv

Description: A variant of opnneir with different dummy variables. (Contributed by Zhi Wang, 31-Aug-2024)

	Ref	Expression
Hypotheses	opnneir.1	⊢ ( 𝜑 → 𝐽 ∈ Top )
	opnneirv.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	opnneirv	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐽 ( 𝑆 ⊆ 𝑥 ∧ 𝜓 ) → ∃ 𝑦 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		opnneir.1	⊢ ( 𝜑 → 𝐽 ∈ Top )
2		opnneirv.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
3	2	opnneilem	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐽 ( 𝑆 ⊆ 𝑥 ∧ 𝜓 ) ↔ ∃ 𝑦 ∈ 𝐽 ( 𝑆 ⊆ 𝑦 ∧ 𝜒 ) ) )
4	1	opnneir	⊢ ( 𝜑 → ( ∃ 𝑦 ∈ 𝐽 ( 𝑆 ⊆ 𝑦 ∧ 𝜒 ) → ∃ 𝑦 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) 𝜒 ) )
5	3 4	sylbid	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐽 ( 𝑆 ⊆ 𝑥 ∧ 𝜓 ) → ∃ 𝑦 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) 𝜒 ) )