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Theorem nfcsb1d 2959
Description: Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
Hypothesis
Ref Expression
nfcsb1d.1 (𝜑𝑥𝐴)
Assertion
Ref Expression
nfcsb1d (𝜑𝑥𝐴 / 𝑥𝐵)

Proof of Theorem nfcsb1d
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-csb 2932 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 nfv 1466 . . 3 𝑦𝜑
3 nfcsb1d.1 . . . 4 (𝜑𝑥𝐴)
43nfsbc1d 2854 . . 3 (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝑦𝐵)
52, 4nfabd 2247 . 2 (𝜑𝑥{𝑦[𝐴 / 𝑥]𝑦𝐵})
61, 5nfcxfrd 2226 1 (𝜑𝑥𝐴 / 𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1438  {cab 2074  wnfc 2215  [wsbc 2838  csb 2931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-sbc 2839  df-csb 2932
This theorem is referenced by:  nfcsb1  2960
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