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Theorem nfiotadxy 4900
Description: Deduction version of nfiotaxy 4901. (Contributed by Jim Kingdon, 21-Dec-2018.)
Hypotheses
Ref Expression
nfiotadxy.1 𝑦𝜑
nfiotadxy.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfiotadxy (𝜑𝑥(℩𝑦𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem nfiotadxy
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfiota2 4898 . 2 (℩𝑦𝜓) = {𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)}
2 nfv 1462 . . . 4 𝑧𝜑
3 nfiotadxy.1 . . . . 5 𝑦𝜑
4 nfiotadxy.2 . . . . . 6 (𝜑 → Ⅎ𝑥𝜓)
5 nfcv 2220 . . . . . . . 8 𝑥𝑦
6 nfcv 2220 . . . . . . . 8 𝑥𝑧
75, 6nfeq 2227 . . . . . . 7 𝑥 𝑦 = 𝑧
87a1i 9 . . . . . 6 (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
94, 8nfbid 1521 . . . . 5 (𝜑 → Ⅎ𝑥(𝜓𝑦 = 𝑧))
103, 9nfald 1684 . . . 4 (𝜑 → Ⅎ𝑥𝑦(𝜓𝑦 = 𝑧))
112, 10nfabd 2238 . . 3 (𝜑𝑥{𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)})
1211nfunid 3616 . 2 (𝜑𝑥 {𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)})
131, 12nfcxfrd 2218 1 (𝜑𝑥(℩𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1283   = wceq 1285  wnf 1390  {cab 2068  wnfc 2207   cuni 3609  cio 4895
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-sn 3412  df-uni 3610  df-iota 4897
This theorem is referenced by:  nfiotaxy  4901  nfriotadxy  5507
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