Theorem raln

Description: Restricted universally quantified negation expressed as a universally quantified negation. (Contributed by BJ, 16-Jul-2021)

	Ref	Expression
Assertion	raln	⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀ 𝑥 ¬ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ¬ 𝜑 ) )
2		imnang	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ¬ 𝜑 ) ↔ ∀ 𝑥 ¬ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
3	1 2	bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀ 𝑥 ¬ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )