Theorem dfnf5 3952

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 1705	. . . 4 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2	1	imbi1i 339	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥𝜑))
3		pm4.64 387	. . 3 ⊢ ((¬ ∀𝑥 ¬ 𝜑 → ∀𝑥𝜑) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
4	2, 3	bitri 264	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
5		df-nf 1710	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
6		ab0 3951	. . 3 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
7		abv 3206	. . 3 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)
8	6, 7	orbi12i 543	. 2 ⊢ (({𝑥 ∣ 𝜑} = ∅ ∨ {𝑥 ∣ 𝜑} = V) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
9	4, 5, 8	3bitr4i 292	1 ⊢ (Ⅎ𝑥𝜑 ↔ ({𝑥 ∣ 𝜑} = ∅ ∨ {𝑥 ∣ 𝜑} = V))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383  ∀wal 1481   = wceq 1483  ∃wex 1704  Ⅎwnf 1708  {cab 2608  Vcvv 3200  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  ab0orv  3953