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Theorem nfcsbd 3832
 Description: Deduction version of nfcsb 3834. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.)
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 3808 . 2 𝐴 / 𝑦𝐵 = {𝑧[𝐴 / 𝑦]𝑧𝐵}
2 nfv 1915 . . 3 𝑧𝜑
3 nfcsbd.1 . . . 4 𝑦𝜑
4 nfcsbd.2 . . . 4 (𝜑𝑥𝐴)
5 nfcsbd.3 . . . . 5 (𝜑𝑥𝐵)
65nfcrd 2908 . . . 4 (𝜑 → Ⅎ𝑥 𝑧𝐵)
73, 4, 6nfsbcd 3722 . . 3 (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝑧𝐵)
82, 7nfabd 2941 . 2 (𝜑𝑥{𝑧[𝐴 / 𝑦]𝑧𝐵})
91, 8nfcxfrd 2918 1 (𝜑𝑥𝐴 / 𝑦𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  Ⅎwnf 1785   ∈ wcel 2111  {cab 2735  Ⅎwnfc 2899  [wsbc 3698  ⦋csb 3807 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-sbc 3699  df-csb 3808 This theorem is referenced by:  nfcsb  3834
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