MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfsbc1d Structured version   Visualization version   GIF version

Theorem nfsbc1d 3793
Description: Deduction version of nfsbc1 3794. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypothesis
Ref Expression
nfsbc1d.2 (𝜑𝑥𝐴)
Assertion
Ref Expression
nfsbc1d (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓)

Proof of Theorem nfsbc1d
StepHypRef Expression
1 df-sbc 3776 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ {𝑥𝜓})
2 nfsbc1d.2 . . 3 (𝜑𝑥𝐴)
3 nfab1 2894 . . . 4 𝑥{𝑥𝜓}
43a1i 11 . . 3 (𝜑𝑥{𝑥𝜓})
52, 4nfeld 2904 . 2 (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑥𝜓})
61, 5nfxfrd 1849 1 (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1778  wcel 2099  {cab 2703  wnfc 2876  [wsbc 3775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-sbc 3776
This theorem is referenced by:  nfsbc1  3794  nfcsb1d  3914
  Copyright terms: Public domain W3C validator