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Theorem nfriotadxy 5852
Description: Deduction version of nfriota 5853. (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 5844 . 2 (𝑦𝐴 𝜓) = (℩𝑦(𝑦𝐴𝜓))
2 nfriotadxy.1 . . 3 𝑦𝜑
3 nfcv 2329 . . . . . 6 𝑥𝑦
43a1i 9 . . . . 5 (𝜑𝑥𝑦)
5 nfriotadxy.3 . . . . 5 (𝜑𝑥𝐴)
64, 5nfeld 2345 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
7 nfriotadxy.2 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
86, 7nfand 1578 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
92, 8nfiotadw 5193 . 2 (𝜑𝑥(℩𝑦(𝑦𝐴𝜓)))
101, 9nfcxfrd 2327 1 (𝜑𝑥(𝑦𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wnf 1470  wcel 2158  wnfc 2316  cio 5188  crio 5843
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-sn 3610  df-uni 3822  df-iota 5190  df-riota 5844
This theorem is referenced by:  nfriota  5853
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