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| Mirrors > Home > ILE Home > Th. List > nfreudxy | GIF version | ||
| Description: Not-free deduction for restricted uniqueness. This is a version where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.) |
| Ref | Expression |
|---|---|
| nfreudxy.1 | ⊢ Ⅎ𝑦𝜑 |
| nfreudxy.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
| nfreudxy.3 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
| Ref | Expression |
|---|---|
| nfreudxy | ⊢ (𝜑 → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfreudxy.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
| 2 | nfcv 2348 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
| 3 | 2 | a1i 9 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦) |
| 4 | nfreudxy.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
| 5 | 3, 4 | nfeld 2364 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
| 6 | nfreudxy.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
| 7 | 5, 6 | nfand 1591 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
| 8 | 1, 7 | nfeud 2070 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
| 9 | df-reu 2491 | . . 3 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
| 10 | 9 | nfbii 1496 | . 2 ⊢ (Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓 ↔ Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
| 11 | 8, 10 | sylibr 134 | 1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ wa 104 Ⅎwnf 1483 ∃!weu 2054 ∈ wcel 2176 Ⅎwnfc 2335 ∃!wreu 2486 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-ext 2187 |
| This theorem depends on definitions: df-bi 117 df-tru 1376 df-nf 1484 df-sb 1786 df-eu 2057 df-cleq 2198 df-clel 2201 df-nfc 2337 df-reu 2491 |
| This theorem is referenced by: nfreuxy 2681 |
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