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Theorem nfrexdxy 2357
Description: Not-free for restricted existential quantification where 𝑥 and 𝑦 are distinct. See nfrexdya 2359 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
Hypotheses
Ref Expression
nfraldxy.2 𝑦𝜑
nfraldxy.3 (𝜑𝑥𝐴)
nfraldxy.4 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfrexdxy (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfrexdxy
StepHypRef Expression
1 df-rex 2312 . 2 (∃𝑦𝐴 𝜓 ↔ ∃𝑦(𝑦𝐴𝜓))
2 nfraldxy.2 . . 3 𝑦𝜑
3 nfcv 2178 . . . . . 6 𝑥𝑦
43a1i 9 . . . . 5 (𝜑𝑥𝑦)
5 nfraldxy.3 . . . . 5 (𝜑𝑥𝐴)
64, 5nfeld 2193 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
7 nfraldxy.4 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
86, 7nfand 1460 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
92, 8nfexd 1644 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝜓))
101, 9nfxfrd 1364 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wnf 1349  wex 1381  wcel 1393  wnfc 2165  wrex 2307
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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312
This theorem is referenced by:  nfrexdya  2359  nfrexxy  2361  nfunid  3587  strcollnft  10083
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