Theorem dfral2

Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997) Allow shortening of rexnal . (Revised by Wolf Lammen, 9-Dec-2019)

	Ref	Expression
Assertion	dfral2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐴 ¬ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		notnotb	⊢ ( 𝜑 ↔ ¬ ¬ 𝜑 )
2	1	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ¬ ¬ 𝜑 )
3		ralnex	⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ ¬ 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐴 ¬ 𝜑 )
4	2 3	bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐴 ¬ 𝜑 )