Theorem rexnal3

Description: Relationship between three restricted universal and existential quantifiers. (Contributed by Thierry Arnoux, 12-Jul-2020)

	Ref	Expression
Assertion	rexnal3	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		rexnal	⊢ ( ∃ 𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∀ 𝑧 ∈ 𝐶 𝜑 )
2	1	2rexbii	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ¬ ∀ 𝑧 ∈ 𝐶 𝜑 )
3		rexnal2	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ¬ ∀ 𝑧 ∈ 𝐶 𝜑 ↔ ¬ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑 )
4	2 3	bitri	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑 )