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Theorem nf3 1882
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 1881 . 2 (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
2 alnex 1877 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
32orbi2i 937 . 2 ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
41, 3bitr4i 270 1 (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198  wo 874  wal 1651  wex 1875  wnf 1879
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 199  df-or 875  df-ex 1876  df-nf 1880
This theorem is referenced by:  nf4  1883  nfnbi  1951  nfnbiOLD  1952  nfim1  2231  dfnf5  4151
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