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Theorem nfreudxy 2483
 Description: Not-free deduction for restricted uniqueness. This is a version where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
Hypotheses
Ref Expression
nfreudxy.1 𝑦𝜑
nfreudxy.2 (𝜑𝑥𝐴)
nfreudxy.3 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfreudxy (𝜑 → Ⅎ𝑥∃!𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfreudxy
StepHypRef Expression
1 nfreudxy.1 . . 3 𝑦𝜑
2 nfcv 2178 . . . . . 6 𝑥𝑦
32a1i 9 . . . . 5 (𝜑𝑥𝑦)
4 nfreudxy.2 . . . . 5 (𝜑𝑥𝐴)
53, 4nfeld 2193 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
6 nfreudxy.3 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
75, 6nfand 1460 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
81, 7nfeud 1916 . 2 (𝜑 → Ⅎ𝑥∃!𝑦(𝑦𝐴𝜓))
9 df-reu 2313 . . 3 (∃!𝑦𝐴 𝜓 ↔ ∃!𝑦(𝑦𝐴𝜓))
109nfbii 1362 . 2 (Ⅎ𝑥∃!𝑦𝐴 𝜓 ↔ Ⅎ𝑥∃!𝑦(𝑦𝐴𝜓))
118, 10sylibr 137 1 (𝜑 → Ⅎ𝑥∃!𝑦𝐴 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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