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Theorem nfriotadxy 5504
Description: Deduction version of nfriota 5505. (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 5496 . 2 (𝑦𝐴 𝜓) = (℩𝑦(𝑦𝐴𝜓))
2 nfriotadxy.1 . . 3 𝑦𝜑
3 nfcv 2194 . . . . . 6 𝑥𝑦
43a1i 9 . . . . 5 (𝜑𝑥𝑦)
5 nfriotadxy.3 . . . . 5 (𝜑𝑥𝐴)
64, 5nfeld 2209 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
7 nfriotadxy.2 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
86, 7nfand 1476 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
92, 8nfiotadxy 4898 . 2 (𝜑𝑥(℩𝑦(𝑦𝐴𝜓)))
101, 9nfcxfrd 2192 1 (𝜑𝑥(𝑦𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wnf 1365  wcel 1409  wnfc 2181  cio 4893  crio 5495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-sn 3409  df-uni 3609  df-iota 4895  df-riota 5496
This theorem is referenced by:  nfriota  5505
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