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Theorem nfcsbd 3924
Description: Deduction version of nfcsb 3926. Usage of this theorem is discouraged because it depends on ax-13 2377. (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 3900 . 2 𝐴 / 𝑦𝐵 = {𝑧[𝐴 / 𝑦]𝑧𝐵}
2 nfv 1914 . . 3 𝑧𝜑
3 nfcsbd.1 . . . 4 𝑦𝜑
4 nfcsbd.2 . . . 4 (𝜑𝑥𝐴)
5 nfcsbd.3 . . . . 5 (𝜑𝑥𝐵)
65nfcrd 2899 . . . 4 (𝜑 → Ⅎ𝑥 𝑧𝐵)
73, 4, 6nfsbcd 3812 . . 3 (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝑧𝐵)
82, 7nfabd 2928 . 2 (𝜑𝑥{𝑧[𝐴 / 𝑦]𝑧𝐵})
91, 8nfcxfrd 2904 1 (𝜑𝑥𝐴 / 𝑦𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1783  wcel 2108  {cab 2714  wnfc 2890  [wsbc 3788  csb 3899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-sbc 3789  df-csb 3900
This theorem is referenced by:  nfcsb  3926
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