Theorem alexn

Description: A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993)

	Ref	Expression
Assertion	alexn	⊢ ( ∀ 𝑥 ∃ 𝑦 ¬ 𝜑 ↔ ¬ ∃ 𝑥 ∀ 𝑦 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		exnal	⊢ ( ∃ 𝑦 ¬ 𝜑 ↔ ¬ ∀ 𝑦 𝜑 )
2	1	albii	⊢ ( ∀ 𝑥 ∃ 𝑦 ¬ 𝜑 ↔ ∀ 𝑥 ¬ ∀ 𝑦 𝜑 )
3		alnex	⊢ ( ∀ 𝑥 ¬ ∀ 𝑦 𝜑 ↔ ¬ ∃ 𝑥 ∀ 𝑦 𝜑 )
4	2 3	bitri	⊢ ( ∀ 𝑥 ∃ 𝑦 ¬ 𝜑 ↔ ¬ ∃ 𝑥 ∀ 𝑦 𝜑 )