Theorem emptynf 1917

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 1916	. 2 ⊢ (¬ ∃𝑥⊤ → ∀𝑥𝜑)
2		nftht 1800	. 2 ⊢ (∀𝑥𝜑 → Ⅎ𝑥𝜑)
3	1, 2	syl 17	1 ⊢ (¬ ∃𝑥⊤ → Ⅎ𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1541  ⊤wtru 1544  ∃wex 1787  Ⅎwnf 1791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817

This theorem depends on definitions:  df-bi 210  df-tru 1546  df-ex 1788  df-nf 1792

This theorem is referenced by: (None)