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Theorem nfreudxy 2650
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 2319 . . . . . 6 𝑥𝑦
32a1i 9 . . . . 5 (𝜑𝑥𝑦)
4 nfreudxy.2 . . . . 5 (𝜑𝑥𝐴)
53, 4nfeld 2335 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
6 nfreudxy.3 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
75, 6nfand 1568 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
81, 7nfeud 2042 . 2 (𝜑 → Ⅎ𝑥∃!𝑦(𝑦𝐴𝜓))
9 df-reu 2462 . . 3 (∃!𝑦𝐴 𝜓 ↔ ∃!𝑦(𝑦𝐴𝜓))
109nfbii 1473 . 2 (Ⅎ𝑥∃!𝑦𝐴 𝜓 ↔ Ⅎ𝑥∃!𝑦(𝑦𝐴𝜓))
118, 10sylibr 134 1 (𝜑 → Ⅎ𝑥∃!𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wnf 1460  ∃!weu 2026  wcel 2148  wnfc 2306  ∃!wreu 2457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-cleq 2170  df-clel 2173  df-nfc 2308  df-reu 2462
This theorem is referenced by:  nfreuxy  2651
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