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Theorem csbie 3930
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019.) Reduce axiom usage. (Revised by Gino Giotto, 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 3895 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 csbie.1 . . . 4 𝐴 ∈ V
3 csbie.2 . . . . 5 (𝑥 = 𝐴𝐵 = 𝐶)
43eleq2d 2820 . . . 4 (𝑥 = 𝐴 → (𝑦𝐵𝑦𝐶))
52, 4sbcie 3821 . . 3 ([𝐴 / 𝑥]𝑦𝐵𝑦𝐶)
65abbii 2803 . 2 {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦𝑦𝐶}
7 abid2 2872 . 2 {𝑦𝑦𝐶} = 𝐶
81, 6, 73eqtri 2765 1 𝐴 / 𝑥𝐵 = 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {cab 2710  Vcvv 3475  [wsbc 3778  csb 3894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-sbc 3779  df-csb 3895
This theorem is referenced by:  pofun  5607  eqerlem  8737  mptnn0fsuppd  13963  fsum  15666  fsumcnv  15719  fsumshftm  15727  fsum0diag2  15729  fprod  15885  fprodcnv  15927  bpolyval  15993  ruclem1  16174  odfval  19400  odval  19402  psrass1lemOLD  21493  psrass1lem  21496  selvval  21681  mamufval  21887  pm2mpval  22297  isibl  25283  dfitg  25287  dvfsumlem2  25544  fsumdvdsmul  26699  precsexlem3  27658  disjxpin  31850  gg-dvfsumlem2  35214  poimirlem1  36537  poimirlem5  36541  poimirlem15  36551  poimirlem16  36552  poimirlem17  36553  poimirlem19  36555  poimirlem20  36556  poimirlem22  36558  poimirlem24  36560  poimirlem28  36564  evlselv  41207  fphpd  41602  monotuz  41728  oddcomabszz  41731  fnwe2val  41839  fnwe2lem1  41840
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