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| Mirrors > Home > MPE Home > Th. List > nfreuw | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3435 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 30-Oct-2010.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.) Avoid ax-9 2118, ax-ext 2708. (Revised by Wolf Lammen, 21-Nov-2024.) |
| Ref | Expression |
|---|---|
| nfrmow.1 | ⊢ Ⅎ𝑥𝐴 |
| nfrmow.2 | ⊢ Ⅎ𝑥𝜑 |
| Ref | Expression |
|---|---|
| nfreuw | ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-reu 3381 | . 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)) | |
| 2 | nfrmow.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
| 3 | 2 | nfcri 2897 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
| 4 | nfrmow.2 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
| 5 | 3, 4 | nfan 1899 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) |
| 6 | 5 | nfeuw 2593 | . 2 ⊢ Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑) |
| 7 | 1, 6 | nfxfr 1853 | 1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑 |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 Ⅎwnf 1783 ∈ wcel 2108 ∃!weu 2568 Ⅎwnfc 2890 ∃!wreu 3378 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-10 2141 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-mo 2540 df-eu 2569 df-clel 2816 df-nfc 2892 df-reu 3381 |
| This theorem is referenced by: sbcreu 3876 reuccatpfxs1 14785 2reu7 47123 2reu8 47124 |
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