Theorem emptyal

Description: On the empty domain, any universally quantified formula is true. (Contributed by Wolf Lammen, 12-Mar-2023)

	Ref	Expression
Assertion	emptyal	⊢ ( ¬ ∃ 𝑥 ⊤ → ∀ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		emptyex	⊢ ( ¬ ∃ 𝑥 ⊤ → ¬ ∃ 𝑥 ¬ 𝜑 )
2		alex	⊢ ( ∀ 𝑥 𝜑 ↔ ¬ ∃ 𝑥 ¬ 𝜑 )
3	1 2	sylibr	⊢ ( ¬ ∃ 𝑥 ⊤ → ∀ 𝑥 𝜑 )