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Theorem nfrexdxy 2443
Description: Not-free for restricted existential quantification where 𝑥 and 𝑦 are distinct. See nfrexdya 2445 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 2397 . 2 (∃𝑦𝐴 𝜓 ↔ ∃𝑦(𝑦𝐴𝜓))
2 nfraldxy.2 . . 3 𝑦𝜑
3 nfcv 2256 . . . . . 6 𝑥𝑦
43a1i 9 . . . . 5 (𝜑𝑥𝑦)
5 nfraldxy.3 . . . . 5 (𝜑𝑥𝐴)
64, 5nfeld 2272 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
7 nfraldxy.4 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
86, 7nfand 1530 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
92, 8nfexd 1717 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝜓))
101, 9nfxfrd 1434 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wnf 1419  wex 1451  wcel 1463  wnfc 2243  wrex 2392
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-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397
This theorem is referenced by:  nfrexdya  2445  nfrexxy  2447  nfunid  3711  strcollnft  13016
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