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 ↔ ({𝑥 ∣ 𝜑} = ∅ ∨ {𝑥 ∣ 𝜑} = V))

Proof of Theorem dfnf5

Step	Hyp	Ref	Expression
1		df-ex 1696	. . . 4 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2	1	imbi1i 338	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥𝜑))
3		pm4.64 386	. . 3 ⊢ ((¬ ∀𝑥 ¬ 𝜑 → ∀𝑥𝜑) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
4	2, 3	bitri 263	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
5		df-nf 1701	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
6		ab0 3905	. . 3 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
7		abv 3179	. . 3 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)
8	6, 7	orbi12i 542	. 2 ⊢ (({𝑥 ∣ 𝜑} = ∅ ∨ {𝑥 ∣ 𝜑} = V) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
9	4, 5, 8	3bitr4i 291	1 ⊢ (Ⅎ𝑥𝜑 ↔ ({𝑥 ∣ 𝜑} = ∅ ∨ {𝑥 ∣ 𝜑} = V))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382  ∀wal 1473   = wceq 1475  ∃wex 1695  Ⅎwnf 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