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Theorem nfsbc1d 2856
Description: Deduction version of nfsbc1 2857. (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 2841 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ {𝑥𝜓})
2 nfsbc1d.2 . . 3 (𝜑𝑥𝐴)
3 nfab1 2230 . . . 4 𝑥{𝑥𝜓}
43a1i 9 . . 3 (𝜑𝑥{𝑥𝜓})
52, 4nfeld 2244 . 2 (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑥𝜓})
61, 5nfxfrd 1409 1 (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1394  wcel 1438  {cab 2074  wnfc 2215  [wsbc 2840
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-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-sbc 2841
This theorem is referenced by:  nfsbc1  2857  nfcsb1d  2961
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