ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfriotadxy GIF version

Theorem nfriotadxy 5855
Description: Deduction version of nfriota 5856. (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 5847 . 2 (𝑦𝐴 𝜓) = (℩𝑦(𝑦𝐴𝜓))
2 nfriotadxy.1 . . 3 𝑦𝜑
3 nfcv 2332 . . . . . 6 𝑥𝑦
43a1i 9 . . . . 5 (𝜑𝑥𝑦)
5 nfriotadxy.3 . . . . 5 (𝜑𝑥𝐴)
64, 5nfeld 2348 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
7 nfriotadxy.2 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
86, 7nfand 1579 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
92, 8nfiotadw 5196 . 2 (𝜑𝑥(℩𝑦(𝑦𝐴𝜓)))
101, 9nfcxfrd 2330 1 (𝜑𝑥(𝑦𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wnf 1471  wcel 2160  wnfc 2319  cio 5191  crio 5846
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-sn 3613  df-uni 3825  df-iota 5193  df-riota 5847
This theorem is referenced by:  nfriota  5856
  Copyright terms: Public domain W3C validator