Theorem nf2 1793

Description: Alternate definition of nonfreeness. (Contributed by BJ, 16-Sep-2021.)

Assertion

Ref	Expression
nf2	⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))

Proof of Theorem nf2

Step	Hyp	Ref	Expression
1		df-nf 1792	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		imor 860	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑))
3		orcom 877	. 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
4	1, 2, 3	3bitri 299	1 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∨ wo 854  ∀wal 1546  ∃wex 1787  Ⅎwnf 1791

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 855  df-nf 1792

This theorem is referenced by:  nf3  1794