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Theorem nfcsbd 3516
 Description: Deduction version of nfcsb 3517. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
nfcsbd.1 𝑦𝜑
nfcsbd.2 (𝜑𝑥𝐴)
nfcsbd.3 (𝜑𝑥𝐵)
Assertion
Ref Expression
nfcsbd (𝜑𝑥𝐴 / 𝑦𝐵)

Proof of Theorem nfcsbd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-csb 3500 . 2 𝐴 / 𝑦𝐵 = {𝑧[𝐴 / 𝑦]𝑧𝐵}
2 nfv 1830 . . 3 𝑧𝜑
3 nfcsbd.1 . . . 4 𝑦𝜑
4 nfcsbd.2 . . . 4 (𝜑𝑥𝐴)
5 nfcsbd.3 . . . . 5 (𝜑𝑥𝐵)
65nfcrd 2757 . . . 4 (𝜑 → Ⅎ𝑥 𝑧𝐵)
73, 4, 6nfsbcd 3423 . . 3 (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝑧𝐵)
82, 7nfabd 2771 . 2 (𝜑𝑥{𝑧[𝐴 / 𝑦]𝑧𝐵})
91, 8nfcxfrd 2750 1 (𝜑𝑥𝐴 / 𝑦𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  Ⅎwnf 1699   ∈ wcel 1977  {cab 2596  Ⅎwnfc 2738  [wsbc 3402  ⦋csb 3499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-sbc 3403  df-csb 3500 This theorem is referenced by:  nfcsb  3517
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