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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 2312 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
3 | 2 | a1i 9 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦) |
4 | nfreudxy.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | 3, 4 | nfeld 2328 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
6 | nfreudxy.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
7 | 5, 6 | nfand 1561 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
8 | 1, 7 | nfeud 2035 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
9 | df-reu 2455 | . . 3 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
10 | 9 | nfbii 1466 | . 2 ⊢ (Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓 ↔ Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
11 | 8, 10 | sylibr 133 | 1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 Ⅎwnf 1453 ∃!weu 2019 ∈ wcel 2141 Ⅎwnfc 2299 ∃!wreu 2450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-cleq 2163 df-clel 2166 df-nfc 2301 df-reu 2455 |
This theorem is referenced by: nfreuxy 2644 |
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