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Theorem nfsbc1d 3809
Description: Deduction version of nfsbc1 3810. (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 3792 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ {𝑥𝜓})
2 nfsbc1d.2 . . 3 (𝜑𝑥𝐴)
3 nfab1 2905 . . . 4 𝑥{𝑥𝜓}
43a1i 11 . . 3 (𝜑𝑥{𝑥𝜓})
52, 4nfeld 2915 . 2 (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑥𝜓})
61, 5nfxfrd 1851 1 (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1780  wcel 2106  {cab 2712  wnfc 2888  [wsbc 3791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-sbc 3792
This theorem is referenced by:  nfsbc1  3810  nfcsb1d  3931
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