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Theorem nfreudxy 2500
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 2194 . . . . . 6 𝑥𝑦
32a1i 9 . . . . 5 (𝜑𝑥𝑦)
4 nfreudxy.2 . . . . 5 (𝜑𝑥𝐴)
53, 4nfeld 2209 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
6 nfreudxy.3 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
75, 6nfand 1476 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
81, 7nfeud 1932 . 2 (𝜑 → Ⅎ𝑥∃!𝑦(𝑦𝐴𝜓))
9 df-reu 2330 . . 3 (∃!𝑦𝐴 𝜓 ↔ ∃!𝑦(𝑦𝐴𝜓))
109nfbii 1378 . 2 (Ⅎ𝑥∃!𝑦𝐴 𝜓 ↔ Ⅎ𝑥∃!𝑦(𝑦𝐴𝜓))
118, 10sylibr 141 1 (𝜑 → Ⅎ𝑥∃!𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wnf 1365  wcel 1409  ∃!weu 1916  wnfc 2181  ∃!wreu 2325
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-cleq 2049  df-clel 2052  df-nfc 2183  df-reu 2330
This theorem is referenced by:  nfreuxy  2501
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