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Theorem nfsbcd 3793
Description: Deduction version of nfsbc 3794. (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 3770 . 2 ([𝐴 / 𝑦]𝜓𝐴 ∈ {𝑦𝜓})
2 nfsbcd.2 . . 3 (𝜑𝑥𝐴)
3 nfsbcd.1 . . . 4 𝑦𝜑
4 nfsbcd.3 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
53, 4nfabd 2998 . . 3 (𝜑𝑥{𝑦𝜓})
62, 5nfeld 2986 . 2 (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦𝜓})
71, 6nfxfrd 1845 1 (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1775  wcel 2105  {cab 2796  wnfc 2958  [wsbc 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-sbc 3770
This theorem is referenced by:  nfsbc  3794  nfcsbd  3905  sbcnestgf  4372
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