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Theorem nfriotadxy 5814
Description: Deduction version of nfriota 5815. (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 5806 . 2 (𝑦𝐴 𝜓) = (℩𝑦(𝑦𝐴𝜓))
2 nfriotadxy.1 . . 3 𝑦𝜑
3 nfcv 2312 . . . . . 6 𝑥𝑦
43a1i 9 . . . . 5 (𝜑𝑥𝑦)
5 nfriotadxy.3 . . . . 5 (𝜑𝑥𝐴)
64, 5nfeld 2328 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
7 nfriotadxy.2 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
86, 7nfand 1561 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
92, 8nfiotadw 5161 . 2 (𝜑𝑥(℩𝑦(𝑦𝐴𝜓)))
101, 9nfcxfrd 2310 1 (𝜑𝑥(𝑦𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wnf 1453  wcel 2141  wnfc 2299  cio 5156  crio 5805
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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-sn 3587  df-uni 3795  df-iota 5158  df-riota 5806
This theorem is referenced by:  nfriota  5815
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