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Theorem nfiotadxy 4813
Description: Deduction version of nfiotaxy 4814. (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 4811 . 2 (℩yψ) = {zy(ψy = z)}
2 nfv 1418 . . . 4 zφ
3 nfiotadxy.1 . . . . 5 yφ
4 nfiotadxy.2 . . . . . 6 (φ → Ⅎxψ)
5 nfcv 2175 . . . . . . . 8 xy
6 nfcv 2175 . . . . . . . 8 xz
75, 6nfeq 2182 . . . . . . 7 x y = z
87a1i 9 . . . . . 6 (φ → Ⅎx y = z)
94, 8nfbid 1477 . . . . 5 (φ → Ⅎx(ψy = z))
103, 9nfald 1640 . . . 4 (φ → Ⅎxy(ψy = z))
112, 10nfabd 2193 . . 3 (φx{zy(ψy = z)})
1211nfunid 3578 . 2 (φx {zy(ψy = z)})
131, 12nfcxfrd 2173 1 (φx(℩yψ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   = wceq 1242  wnf 1346  {cab 2023  wnfc 2162   cuni 3571  cio 4808
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-sn 3373  df-uni 3572  df-iota 4810
This theorem is referenced by:  nfiotaxy  4814  nfriotadxy  5419
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