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Theorem nfsbcd 3795
 Description: Deduction version of nfsbc 3796. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker nfsbcdw 3792 when possible. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.)
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 3772 . 2 ([𝐴 / 𝑦]𝜓𝐴 ∈ {𝑦𝜓})
2 nfsbcd.2 . . 3 (𝜑𝑥𝐴)
3 nfsbcd.1 . . . 4 𝑦𝜑
4 nfsbcd.3 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
53, 4nfabd 3001 . . 3 (𝜑𝑥{𝑦𝜓})
62, 5nfeld 2989 . 2 (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦𝜓})
71, 6nfxfrd 1850 1 (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  Ⅎwnf 1780   ∈ wcel 2110  {cab 2799  Ⅎwnfc 2961  [wsbc 3771 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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-13 2386  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-sbc 3772 This theorem is referenced by:  nfsbc  3796  nfcsbd  3907  sbcnestgf  4374
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