Theorem emptyal 1941

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

Assertion

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

Proof of Theorem emptyal

Step	Hyp	Ref	Expression
1		emptyex 1940	. 2 ⊢ (¬ ∃𝑥⊤ → ¬ ∃𝑥 ¬ 𝜑)
2		alex 1859	. 2 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
3	1, 2	sylibr 237	1 ⊢ (¬ ∃𝑥⊤ → ∀𝑥𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568  ⊤wtru 1571  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-tru 1573  df-ex 1813

This theorem is used by:  emptynf  1942