Theorem nf3 1818

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 1817	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
2		alnex 1813	. . 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 1567  ∃wex 1811  Ⅎwnf 1815

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 1812  df-nf 1816

This theorem is used by:  nf4  1819  nfim1  2235  dfnf5  4330  ab0orv  4331