Theorem emptyex 1939

Description: On the empty domain, any existentially quantified formula is false. (Contributed by Wolf Lammen, 21-Jan-2024.)

Assertion

Ref	Expression
emptyex	⊢ (¬ ∃𝑥⊤ → ¬ ∃𝑥𝜑)

Proof of Theorem emptyex

Step	Hyp	Ref	Expression
1		trud 1579	. . 3 ⊢ (𝜑 → ⊤)
2	1	eximi 1867	. 2 ⊢ (∃𝑥𝜑 → ∃𝑥⊤)
3	2	con3i 155	1 ⊢ (¬ ∃𝑥⊤ → ¬ ∃𝑥𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ⊤wtru 1570  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841

This proof depends on definitions:  df-bi 210  df-tru 1572  df-ex 1812

This theorem is used by:  emptyal  1940