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Theorem dfnf5 4338
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 1789 . 2 (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
2 abv 3455 . . 3 ({𝑥𝜑} = V ↔ ∀𝑥𝜑)
3 ab0 4335 . . 3 ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
42, 3orbi12i 914 . 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 1540   = wceq 1542  wnf 1786  {cab 2710  Vcvv 3444  c0 4283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-v 3446  df-dif 3914  df-nul 4284
This theorem is referenced by:  ab0orvALT  4340
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