Theorem emptynf

Description: On the empty domain, any variable is effectively nonfree in any formula. (Contributed by Wolf Lammen, 12-Mar-2023)

	Ref	Expression
Assertion	emptynf	⊢ ( ¬ ∃ 𝑥 ⊤ → Ⅎ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		emptyal	⊢ ( ¬ ∃ 𝑥 ⊤ → ∀ 𝑥 𝜑 )
2		nftht	⊢ ( ∀ 𝑥 𝜑 → Ⅎ 𝑥 𝜑 )
3	1 2	syl	⊢ ( ¬ ∃ 𝑥 ⊤ → Ⅎ 𝑥 𝜑 )