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 Description: Deduction version of nfiotaxy 4871. (Contributed by Jim Kingdon, 21-Dec-2018.)
Hypotheses
Ref Expression
Assertion
Ref Expression
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfiota2 4868 . 2 (℩𝑦𝜓) = {𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)}
2 nfv 1421 . . . 4 𝑧𝜑
3 nfiotadxy.1 . . . . 5 𝑦𝜑
4 nfiotadxy.2 . . . . . 6 (𝜑 → Ⅎ𝑥𝜓)
5 nfcv 2178 . . . . . . . 8 𝑥𝑦
6 nfcv 2178 . . . . . . . 8 𝑥𝑧
75, 6nfeq 2185 . . . . . . 7 𝑥 𝑦 = 𝑧
87a1i 9 . . . . . 6 (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
94, 8nfbid 1480 . . . . 5 (𝜑 → Ⅎ𝑥(𝜓𝑦 = 𝑧))
103, 9nfald 1643 . . . 4 (𝜑 → Ⅎ𝑥𝑦(𝜓𝑦 = 𝑧))
112, 10nfabd 2196 . . 3 (𝜑𝑥{𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)})
1211nfunid 3587 . 2 (𝜑𝑥 {𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)})
131, 12nfcxfrd 2176 1 (𝜑𝑥(℩𝑦𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241   = wceq 1243  Ⅎwnf 1349  {cab 2026  Ⅎwnfc 2165  ∪ cuni 3580  ℩cio 4865 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 2312  df-sn 3381  df-uni 3581  df-iota 4867 This theorem is referenced by:  nfiotaxy  4871  nfriotadxy  5476
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