Theorem nf4 1784

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 1783	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
2		df-or 844	. 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
3	1, 2	bitri 277	1 ⊢ (Ⅎ𝑥𝜑 ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∨ wo 843  ∀wal 1531  Ⅎwnf 1780

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 209  df-or 844  df-ex 1777  df-nf 1781

This theorem is referenced by: (None)