Theorem rexnal2

Description: Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	rexnal2	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 )

Proof

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