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Theorem nfcsb1d 3705
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 3692 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 nfv 2009 . . 3 𝑦𝜑
3 nfcsb1d.1 . . . 4 (𝜑𝑥𝐴)
43nfsbc1d 3614 . . 3 (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝑦𝐵)
52, 4nfabd 2928 . 2 (𝜑𝑥{𝑦[𝐴 / 𝑥]𝑦𝐵})
61, 5nfcxfrd 2906 1 (𝜑𝑥𝐴 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2155  {cab 2751  wnfc 2894  [wsbc 3596  csb 3691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-sbc 3597  df-csb 3692
This theorem is referenced by:  nfcsb1  3706  riotaeqimp  6826
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