Theorem rexnal

Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997) (Proof shortened by Wolf Lammen, 9-Dec-2019)

	Ref	Expression
Assertion	rexnal	⊢ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀ 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		dfral2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐴 ¬ 𝜑 )
2	1	con2bii	⊢ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀ 𝑥 ∈ 𝐴 𝜑 )