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Mirrors > Home > MPE Home > Th. List > dfnf5 | Structured version Visualization version GIF version |
Description: Characterization of nonfreeness in a formula in terms of its extension. (Contributed by BJ, 19-Mar-2021.) |
Ref | Expression |
---|---|
dfnf5 | ⊢ (Ⅎ𝑥𝜑 ↔ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nf3 1780 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)) | |
2 | abv 3477 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑) | |
3 | ab0 4367 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) | |
4 | 2, 3 | orbi12i 911 | . 2 ⊢ (({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅) ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)) |
5 | 1, 4 | bitr4i 278 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∨ wo 844 ∀wal 1531 = wceq 1533 Ⅎwnf 1777 {cab 2701 Vcvv 3466 ∅c0 4315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-v 3468 df-dif 3944 df-nul 4316 |
This theorem is referenced by: ab0orvALT 4372 |
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