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Mirrors > Home > MPE Home > Th. List > dfnf5 | Structured version Visualization version GIF version |
Description: Characterization of non-freeness 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 1788 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)) | |
2 | abv 3451 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑) | |
3 | ab0 4287 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑) | |
4 | 2, 3 | orbi12i 912 | . 2 ⊢ (({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅) ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)) |
5 | 1, 4 | bitr4i 281 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∨ wo 844 ∀wal 1536 = wceq 1538 Ⅎwnf 1785 {cab 2776 Vcvv 3441 ∅c0 4243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-nul 4244 |
This theorem is referenced by: ab0orv 4289 |
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