Theorem nfnbi 1854

Description: A variable is nonfree in a proposition if and only if it is so in its negation. (Contributed by BJ, 6-May-2019.) (Proof shortened by Wolf Lammen, 6-Oct-2024.)

Assertion

Ref	Expression
nfnbi	⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)

Proof of Theorem nfnbi

Step	Hyp	Ref	Expression
1		exnal 1826	. . 3 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
2	1	imbi1i 349	. 2 ⊢ ((∃𝑥 ¬ 𝜑 → ∀𝑥 ¬ 𝜑) ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
3		df-nf 1783	. 2 ⊢ (Ⅎ𝑥 ¬ 𝜑 ↔ (∃𝑥 ¬ 𝜑 → ∀𝑥 ¬ 𝜑))
4		nf4 1786	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
5	2, 3, 4	3bitr4ri 304	1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1537  ∃wex 1778  Ⅎwnf 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808

This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1779  df-nf 1783

This theorem is referenced by:  nfnt  1855  wl-sb8et  37555