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

Theorem nfsbcd 2844
Description: Deduction version of nfsbc 2845. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
nfsbcd.1 𝑦𝜑
nfsbcd.2 (𝜑𝑥𝐴)
nfsbcd.3 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfsbcd (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)

Proof of Theorem nfsbcd
StepHypRef Expression
1 df-sbc 2826 . 2 ([𝐴 / 𝑦]𝜓𝐴 ∈ {𝑦𝜓})
2 nfsbcd.2 . . 3 (𝜑𝑥𝐴)
3 nfsbcd.1 . . . 4 𝑦𝜑
4 nfsbcd.3 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
53, 4nfabd 2241 . . 3 (𝜑𝑥{𝑦𝜓})
62, 5nfeld 2238 . 2 (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦𝜓})
71, 6nfxfrd 1405 1 (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1390  wcel 1434  {cab 2069  wnfc 2210  [wsbc 2825
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 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-sbc 2826
This theorem is referenced by:  nfsbc  2845  nfcsbd  2949  sbcnestgf  2963
  Copyright terms: Public domain W3C validator