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Theorem dfnf5 4087
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 1846 . . . 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 264 . 2 ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
5 df-nf 1851 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
6 ab0 4086 . . 3 ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
7 abv 3338 . . 3 ({𝑥𝜑} = V ↔ ∀𝑥𝜑)
86, 7orbi12i 544 . 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 382  wal 1622   = wceq 1624  wex 1845  wnf 1849  {cab 2738  Vcvv 3332  c0 4050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-v 3334  df-dif 3710  df-nul 4051
This theorem is referenced by:  ab0orv  4088
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