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Theorem dfnf5 4099
Description: Characterization of non-freeness in a formula in terms of its extension. (Contributed by BJ, 19-Mar-2021.)
Assertion
Ref Expression
dfnf5 (Ⅎ𝑥𝜑 ↔ ({𝑥𝜑} = ∅ ∨ {𝑥𝜑} = V))

Proof of Theorem dfnf5
StepHypRef Expression
1 df-ex 1853 . . . 4 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
21imbi1i 338 . . 3 ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥𝜑))
3 pm4.64 838 . . 3 ((¬ ∀𝑥 ¬ 𝜑 → ∀𝑥𝜑) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
42, 3bitri 264 . 2 ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
5 df-nf 1858 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
6 ab0 4098 . . 3 ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
7 abv 3358 . . 3 ({𝑥𝜑} = V ↔ ∀𝑥𝜑)
86, 7orbi12i 900 . 2 (({𝑥𝜑} = ∅ ∨ {𝑥𝜑} = V) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
94, 5, 83bitr4i 292 1 (Ⅎ𝑥𝜑 ↔ ({𝑥𝜑} = ∅ ∨ {𝑥𝜑} = V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 836  wal 1629   = wceq 1631  wex 1852  wnf 1856  {cab 2757  Vcvv 3351  c0 4063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-nul 4064
This theorem is referenced by:  ab0orv  4100
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