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

Theorem nfsbcd 3423
 Description: Deduction version of nfsbc 3424. (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 3403 . 2 ([𝐴 / 𝑦]𝜓𝐴 ∈ {𝑦𝜓})
2 nfsbcd.2 . . 3 (𝜑𝑥𝐴)
3 nfsbcd.1 . . . 4 𝑦𝜑
4 nfsbcd.3 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
53, 4nfabd 2771 . . 3 (𝜑𝑥{𝑦𝜓})
62, 5nfeld 2759 . 2 (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦𝜓})
71, 6nfxfrd 1772 1 (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  Ⅎwnf 1699   ∈ wcel 1977  {cab 2596  Ⅎwnfc 2738  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-sbc 3403 This theorem is referenced by:  nfsbc  3424  nfcsbd  3516  sbcnestgf  3947
 Copyright terms: Public domain W3C validator