Theorem emptynf 1942

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

Assertion

Ref	Expression
emptynf	⊢ (¬ ∃𝑥⊤ → Ⅎ𝑥𝜑)

Proof of Theorem emptynf

Step	Hyp	Ref	Expression
1		emptyal 1941	. 2 ⊢ (¬ ∃𝑥⊤ → ∀𝑥𝜑)
2		nftht 1825	. 2 ⊢ (∀𝑥𝜑 → Ⅎ𝑥𝜑)
3	1, 2	syl 18	1 ⊢ (¬ ∃𝑥⊤ → Ⅎ𝑥𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568  ⊤wtru 1571  ∃wex 1812  Ⅎwnf 1816

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  df-nf 1817

This theorem is used by: (None)