MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfabd Structured version   Visualization version   GIF version

Theorem nfabd 2954
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-9 2116 and ax-ext 2754. (Revised by Wolf Lammen, 23-May-2023.)
Hypotheses
Ref Expression
nfabd.1 𝑦𝜑
nfabd.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfabd (𝜑𝑥{𝑦𝜓})

Proof of Theorem nfabd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1957 . 2 𝑧𝜑
2 df-clab 2764 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
3 nfabd.1 . . . 4 𝑦𝜑
4 nfabd.2 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
53, 4nfsbd 2522 . . 3 (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
62, 5nfxfrd 1898 . 2 (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦𝜓})
71, 6nfcd 2926 1 (𝜑𝑥{𝑦𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1827  [wsb 2011  wcel 2107  {cab 2763  wnfc 2919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-nfc 2921
This theorem is referenced by:  nfabd2  2955  nfsbcd  3673  nfcsb1d  3765  nfcsbd  3768  nfifd  4335  nfunid  4680  nfiotad  6104  nfintd  43539  nfiund  43540
  Copyright terms: Public domain W3C validator