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Theorem nf3 1825
Description: Alternate definition of non-freeness. (Contributed by BJ, 16-Sep-2021.)
Assertion
Ref Expression
nf3 (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))

Proof of Theorem nf3
StepHypRef Expression
1 nf2 1824 . 2 (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
2 alnex 1819 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
32orbi2i 542 . 2 ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
41, 3bitr4i 267 1 (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 382  wal 1594  wex 1817  wnf 1821
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 197  df-or 384  df-ex 1818  df-nf 1823
This theorem is referenced by:  nf4  1826  nfntht2  1833  nfnbi  1894  nfntOLDOLD  1896  nfim1  2178
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