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Theorem nfsbcd 3438
Description: Deduction version of nfsbc 3439. (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 3418 . 2 ([𝐴 / 𝑦]𝜓𝐴 ∈ {𝑦𝜓})
2 nfsbcd.2 . . 3 (𝜑𝑥𝐴)
3 nfsbcd.1 . . . 4 𝑦𝜑
4 nfsbcd.3 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
53, 4nfabd 2781 . . 3 (𝜑𝑥{𝑦𝜓})
62, 5nfeld 2769 . 2 (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦𝜓})
71, 6nfxfrd 1777 1 (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1705  wcel 1987  {cab 2607  wnfc 2748  [wsbc 3417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-sbc 3418
This theorem is referenced by:  nfsbc  3439  nfcsbd  3531  sbcnestgf  3967
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