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Theorem nfreudxy 2630
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 2299 . . . . . 6 𝑥𝑦
32a1i 9 . . . . 5 (𝜑𝑥𝑦)
4 nfreudxy.2 . . . . 5 (𝜑𝑥𝐴)
53, 4nfeld 2315 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
6 nfreudxy.3 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
75, 6nfand 1548 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
81, 7nfeud 2022 . 2 (𝜑 → Ⅎ𝑥∃!𝑦(𝑦𝐴𝜓))
9 df-reu 2442 . . 3 (∃!𝑦𝐴 𝜓 ↔ ∃!𝑦(𝑦𝐴𝜓))
109nfbii 1453 . 2 (Ⅎ𝑥∃!𝑦𝐴 𝜓 ↔ Ⅎ𝑥∃!𝑦(𝑦𝐴𝜓))
118, 10sylibr 133 1 (𝜑 → Ⅎ𝑥∃!𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wnf 1440  ∃!weu 2006  wcel 2128  wnfc 2286  ∃!wreu 2437
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-cleq 2150  df-clel 2153  df-nfc 2288  df-reu 2442
This theorem is referenced by:  nfreuxy  2631
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