Theorem ralnex2

Description: Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019) (Proof shortened by Wolf Lammen, 18-May-2023)

	Ref	Expression
Assertion	ralnex2	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ralnex	⊢ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃ 𝑦 ∈ 𝐵 𝜑 )
2	1	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ¬ ∃ 𝑦 ∈ 𝐵 𝜑 )
3		ralnex	⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 )
4	2 3	bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 )