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

Theorem nfiotadxy 4797
Description: Deduction version of nfiotaxy 4798. (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 4795 . 2 (℩yψ) = {zy(ψy = z)}
2 nfv 1402 . . . 4 zφ
3 nfiotadxy.1 . . . . 5 yφ
4 nfiotadxy.2 . . . . . 6 (φ → Ⅎxψ)
5 nfcv 2160 . . . . . . . 8 xy
6 nfcv 2160 . . . . . . . 8 xz
75, 6nfeq 2167 . . . . . . 7 x y = z
87a1i 9 . . . . . 6 (φ → Ⅎx y = z)
94, 8nfbid 1462 . . . . 5 (φ → Ⅎx(ψy = z))
103, 9nfald 1625 . . . 4 (φ → Ⅎxy(ψy = z))
112, 10nfabd 2178 . . 3 (φx{zy(ψy = z)})
1211nfunid 3561 . 2 (φx {zy(ψy = z)})
131, 12nfcxfrd 2158 1 (φx(℩yψ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1226   = wceq 1228  wnf 1329  {cab 2008  wnfc 2147   cuni 3554  cio 4792
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-sn 3356  df-uni 3555  df-iota 4794
This theorem is referenced by:  nfiotaxy  4798  nfriotadxy  5400
  Copyright terms: Public domain W3C validator