Theorem nf4 1713

Description: Alternate definition of non-freeness. This definition uses only primitive symbols. (Contributed by BJ, 16-Sep-2021.)

Assertion

Ref	Expression
nf4	⊢ (Ⅎ𝑥𝜑 ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))

Proof of Theorem nf4

Step	Hyp	Ref	Expression
1		nf3 1712	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
2		df-or 385	. 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
3	1, 2	bitri 264	1 ⊢ (Ⅎ𝑥𝜑 ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383  ∀wal 1481  Ⅎwnf 1708

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by:  nfimdOLDOLD  1824