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Theorem nf3 1783
Description: Alternate definition of nonfreeness. (Contributed by BJ, 16-Sep-2021.)
Assertion
Ref Expression
nf3 (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))

Proof of Theorem nf3
StepHypRef Expression
1 nf2 1782 . 2 (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
2 alnex 1778 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
32orbi2i 912 . 2 ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
41, 3bitr4i 278 1 (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wo 847  wal 1535  wex 1776  wnf 1780
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 848  df-ex 1777  df-nf 1781
This theorem is referenced by:  nf4  1784  nfnbiOLD  1853  nfim1  2197  dfnf5  4388  ab0orv  4389
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