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Theorem csbie 3934
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019.) Reduce axiom usage. (Revised by GG, 15-Oct-2024.)
Hypotheses
Ref Expression
csbie.1 𝐴 ∈ V
csbie.2 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
csbie 𝐴 / 𝑥𝐵 = 𝐶
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem csbie
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-csb 3900 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 csbie.1 . . . 4 𝐴 ∈ V
3 csbie.2 . . . . 5 (𝑥 = 𝐴𝐵 = 𝐶)
43eleq2d 2827 . . . 4 (𝑥 = 𝐴 → (𝑦𝐵𝑦𝐶))
52, 4sbcie 3830 . . 3 ([𝐴 / 𝑥]𝑦𝐵𝑦𝐶)
65abbii 2809 . 2 {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦𝑦𝐶}
7 abid2 2879 . 2 {𝑦𝑦𝐶} = 𝐶
81, 6, 73eqtri 2769 1 𝐴 / 𝑥𝐵 = 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {cab 2714  Vcvv 3480  [wsbc 3788  csb 3899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-sbc 3789  df-csb 3900
This theorem is referenced by:  pofun  5610  eqerlem  8780  mptnn0fsuppd  14039  fsum  15756  fsumcnv  15809  fsumshftm  15817  fsum0diag2  15819  fprod  15977  fprodcnv  16019  bpolyval  16085  ruclem1  16267  odfval  19550  odval  19552  psrass1lem  21952  selvval  22139  mamufval  22396  pm2mpval  22801  isibl  25800  dfitg  25804  dvfsumlem2  26067  dvfsumlem2OLD  26068  fsumdvdsmul  27238  fsumdvdsmulOLD  27240  precsexlem3  28233  disjxpin  32601  poimirlem1  37628  poimirlem5  37632  poimirlem15  37642  poimirlem16  37643  poimirlem17  37644  poimirlem19  37646  poimirlem20  37647  poimirlem22  37649  poimirlem24  37651  poimirlem28  37655  evlselv  42597  fphpd  42827  monotuz  42953  oddcomabszz  42956  fnwe2val  43061  fnwe2lem1  43062  dfswapf2  48967
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