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Theorem nfcsb1d 3062
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 3032 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 nfv 1508 . . 3 𝑦𝜑
3 nfcsb1d.1 . . . 4 (𝜑𝑥𝐴)
43nfsbc1d 2953 . . 3 (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝑦𝐵)
52, 4nfabd 2319 . 2 (𝜑𝑥{𝑦[𝐴 / 𝑥]𝑦𝐵})
61, 5nfcxfrd 2297 1 (𝜑𝑥𝐴 / 𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2128  {cab 2143  wnfc 2286  [wsbc 2937  csb 3031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-sbc 2938  df-csb 3032
This theorem is referenced by:  nfcsb1  3063
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