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Theorem nfcsb1d 3080
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 3050 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 nfv 1521 . . 3 𝑦𝜑
3 nfcsb1d.1 . . . 4 (𝜑𝑥𝐴)
43nfsbc1d 2971 . . 3 (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝑦𝐵)
52, 4nfabd 2332 . 2 (𝜑𝑥{𝑦[𝐴 / 𝑥]𝑦𝐵})
61, 5nfcxfrd 2310 1 (𝜑𝑥𝐴 / 𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2141  {cab 2156  wnfc 2299  [wsbc 2955  csb 3049
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-sbc 2956  df-csb 3050
This theorem is referenced by:  nfcsb1  3081
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