Theorem nf3 1882

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

Assertion

Ref	Expression
nf3	⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))

Proof of Theorem nf3

Step	Hyp	Ref	Expression
1		nf2 1881	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
2		alnex 1877	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
3	2	orbi2i 937	. 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
4	1, 3	bitr4i 270	1 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 198   ∨ wo 874  ∀wal 1651  ∃wex 1875  Ⅎwnf 1879

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 199  df-or 875  df-ex 1876  df-nf 1880

This theorem is referenced by:  nf4  1883  nfnbi  1951  nfnbiOLD  1952  nfim1  2231  dfnf5  4151