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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 2178 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
3 | 2 | a1i 9 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦) |
4 | nfreudxy.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | 3, 4 | nfeld 2193 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
6 | nfreudxy.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
7 | 5, 6 | nfand 1460 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
8 | 1, 7 | nfeud 1916 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
9 | df-reu 2313 | . . 3 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
10 | 9 | nfbii 1362 | . 2 ⊢ (Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓 ↔ Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
11 | 8, 10 | sylibr 137 | 1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 Ⅎwnf 1349 ∈ wcel 1393 ∃!weu 1900 Ⅎwnfc 2165 ∃!wreu 2308 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-cleq 2033 df-clel 2036 df-nfc 2167 df-reu 2313 |
This theorem is referenced by: nfreuxy 2484 |
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