Theorem ralnex

Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997) (Proof shortened by BJ, 16-Jul-2021)

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

Proof

Step	Hyp	Ref	Expression
1		raln	⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀ 𝑥 ¬ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
2		alnex	⊢ ( ∀ 𝑥 ¬ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ¬ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
3		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
4	2 3	xchbinxr	⊢ ( ∀ 𝑥 ¬ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ¬ ∃ 𝑥 ∈ 𝐴 𝜑 )
5	1 4	bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐴 𝜑 )