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Theorem dfnf5 4377
Description: Characterization of nonfreeness in a formula in terms of its extension. (Contributed by BJ, 19-Mar-2021.)
Assertion
Ref Expression
dfnf5 (Ⅎ𝑥𝜑 ↔ ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))

Proof of Theorem dfnf5
StepHypRef Expression
1 nf3 1788 . 2 (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
2 abv 3485 . . 3 ({𝑥𝜑} = V ↔ ∀𝑥𝜑)
3 ab0 4374 . . 3 ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
42, 3orbi12i 913 . 2 (({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅) ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
51, 4bitr4i 277 1 (Ⅎ𝑥𝜑 ↔ ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wo 845  wal 1539   = wceq 1541  wnf 1785  {cab 2709  Vcvv 3474  c0 4322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-v 3476  df-dif 3951  df-nul 4323
This theorem is referenced by:  ab0orvALT  4379
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