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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 1781 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)) | |
2 | abv 3482 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑) | |
3 | ab0 4375 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) | |
4 | 2, 3 | orbi12i 913 | . 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 846 ∀wal 1532 = wceq 1534 Ⅎwnf 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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