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Theorem dfnf5 4378
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 1781 . 2 (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
2 abv 3482 . . 3 ({𝑥𝜑} = V ↔ ∀𝑥𝜑)
3 ab0 4375 . . 3 ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
42, 3orbi12i 913 . 2 (({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅) ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
51, 4bitr4i 278 1 (Ⅎ𝑥𝜑 ↔ ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wo 846  wal 1532   = wceq 1534  wnf 1778  {cab 2705  Vcvv 3471  c0 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-v 3473  df-dif 3950  df-nul 4324
This theorem is referenced by:  ab0orvALT  4380
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