Theorem nf3 1784

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 1783	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
2		alnex 1779	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
3	2	orbi2i 911	. 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
4	1, 3	bitr4i 278	1 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∨ wo 846  ∀wal 1535  ∃wex 1777  Ⅎwnf 1781

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 207  df-or 847  df-ex 1778  df-nf 1782

This theorem is referenced by:  nf4  1785  nfnbiOLD  1854  nfim1  2200  dfnf5  4405  ab0orv  4406