Theorem csbie 3957

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 = 𝐴 → 𝐵 = 𝐶)

Assertion

Ref	Expression
csbie	⊢ ⦋𝐴 / 𝑥⦌𝐵 = 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.

Step	Hyp	Ref	Expression
1		df-csb 3922	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		csbie.1	. . . 4 ⊢ 𝐴 ∈ V
3		csbie.2	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
4	3	eleq2d 2830	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
5	2, 4	sbcie 3848	. . 3 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)
6	5	abbii 2812	. 2 ⊢ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ 𝑦 ∈ 𝐶}
7		abid2 2882	. 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐶} = 𝐶
8	1, 6, 7	3eqtri 2772	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  {cab 2717  Vcvv 3488  [wsbc 3804  ⦋csb 3921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-sbc 3805  df-csb 3922

This theorem is referenced by:  pofun  5626  eqerlem  8798  mptnn0fsuppd  14049  fsum  15768  fsumcnv  15821  fsumshftm  15829  fsum0diag2  15831  fprod  15989  fprodcnv  16031  bpolyval  16097  ruclem1  16279  odfval  19574  odval  19576  psrass1lem  21975  selvval  22162  mamufval  22417  pm2mpval  22822  isibl  25820  dfitg  25824  dvfsumlem2  26087  dvfsumlem2OLD  26088  fsumdvdsmul  27256  fsumdvdsmulOLD  27258  precsexlem3  28251  disjxpin  32610  poimirlem1  37581  poimirlem5  37585  poimirlem15  37595  poimirlem16  37596  poimirlem17  37597  poimirlem19  37599  poimirlem20  37600  poimirlem22  37602  poimirlem24  37604  poimirlem28  37608  evlselv  42542  fphpd  42772  monotuz  42898  oddcomabszz  42901  fnwe2val  43006  fnwe2lem1  43007