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Theorem dfnf5 3906
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 1696 . . . 4 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
21imbi1i 338 . . 3 ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥𝜑))
3 pm4.64 386 . . 3 ((¬ ∀𝑥 ¬ 𝜑 → ∀𝑥𝜑) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
42, 3bitri 263 . 2 ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
5 df-nf 1701 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
6 ab0 3905 . . 3 ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
7 abv 3179 . . 3 ({𝑥𝜑} = V ↔ ∀𝑥𝜑)
86, 7orbi12i 542 . 2 (({𝑥𝜑} = ∅ ∨ {𝑥𝜑} = V) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
94, 5, 83bitr4i 291 1 (Ⅎ𝑥𝜑 ↔ ({𝑥𝜑} = ∅ ∨ {𝑥𝜑} = V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wal 1473   = wceq 1475  wex 1695  wnf 1699  {cab 2596  Vcvv 3173  c0 3874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-nul 3875
This theorem is referenced by:  ab0orv  3907
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