Theorem nf3 1819

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

Assertion

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

Proof of Theorem nf3

Step	Hyp	Ref	Expression
1		nf2 1818	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
2		alnex 1814	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
3	2	orbi2i 926	. 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
4	1, 3	bitr4i 281	1 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∨ wo 861  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by:  nf4  1820  nfim1  2238  dfnf5  4341  ab0orv  4342