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Theorem nfcsb1d 3512
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 3499 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 nfv 1829 . . 3 𝑦𝜑
3 nfcsb1d.1 . . . 4 (𝜑𝑥𝐴)
43nfsbc1d 3419 . . 3 (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝑦𝐵)
52, 4nfabd 2770 . 2 (𝜑𝑥{𝑦[𝐴 / 𝑥]𝑦𝐵})
61, 5nfcxfrd 2749 1 (𝜑𝑥𝐴 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1976  {cab 2595  wnfc 2737  [wsbc 3401  csb 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-sbc 3402  df-csb 3499
This theorem is referenced by:  nfcsb1  3513  riotaeqimp  40144
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