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Theorem nfcsbd 3934
Description: Deduction version of nfcsb 3936. Usage of this theorem is discouraged because it depends on ax-13 2375. (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 3909 . 2 𝐴 / 𝑦𝐵 = {𝑧[𝐴 / 𝑦]𝑧𝐵}
2 nfv 1912 . . 3 𝑧𝜑
3 nfcsbd.1 . . . 4 𝑦𝜑
4 nfcsbd.2 . . . 4 (𝜑𝑥𝐴)
5 nfcsbd.3 . . . . 5 (𝜑𝑥𝐵)
65nfcrd 2897 . . . 4 (𝜑 → Ⅎ𝑥 𝑧𝐵)
73, 4, 6nfsbcd 3815 . . 3 (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝑧𝐵)
82, 7nfabd 2926 . 2 (𝜑𝑥{𝑧[𝐴 / 𝑦]𝑧𝐵})
91, 8nfcxfrd 2902 1 (𝜑𝑥𝐴 / 𝑦𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1780  wcel 2106  {cab 2712  wnfc 2888  [wsbc 3791  csb 3908
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-sbc 3792  df-csb 3909
This theorem is referenced by:  nfcsb  3936
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