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Theorem nfriotadxy 5439
Description: Deduction version of nfriota 5440. (Contributed by Jim Kingdon, 12-Jan-2019.)
Hypotheses
Ref Expression
nfriotadxy.1 𝑦𝜑
nfriotadxy.2 (𝜑 → Ⅎ𝑥𝜓)
nfriotadxy.3 (𝜑𝑥𝐴)
Assertion
Ref Expression
nfriotadxy (𝜑𝑥(𝑦𝐴 𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfriotadxy
StepHypRef Expression
1 df-riota 5431 . 2 (𝑦𝐴 𝜓) = (℩𝑦(𝑦𝐴𝜓))
2 nfriotadxy.1 . . 3 𝑦𝜑
3 nfcv 2178 . . . . . 6 𝑥𝑦
43a1i 9 . . . . 5 (𝜑𝑥𝑦)
5 nfriotadxy.3 . . . . 5 (𝜑𝑥𝐴)
64, 5nfeld 2193 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
7 nfriotadxy.2 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
86, 7nfand 1460 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
92, 8nfiotadxy 4833 . 2 (𝜑𝑥(℩𝑦(𝑦𝐴𝜓)))
101, 9nfcxfrd 2176 1 (𝜑𝑥(𝑦𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wnf 1349  wcel 1393  wnfc 2165  cio 4828  crio 5430
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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2309  df-sn 3378  df-uni 3578  df-iota 4830  df-riota 5431
This theorem is referenced by:  nfriota  5440
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