Theorem csbie 3890

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 3856	. 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 2851	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
5	2, 4	sbcie 3788	. . 3 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)
6	5	abbii 2832	. 2 ⊢ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ 𝑦 ∈ 𝐶}
7		abid2 2902	. 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐶} = 𝐶
8	1, 6, 7	3eqtri 2792	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  {cab 2743  Vcvv 3457  [wsbc 3747  ⦋csb 3855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-sbc 3748  df-csb 3856

This theorem is referenced by:  pofun  5578  eqerlem  8718  mptnn0fsuppd  14025  fsum  15761  fsumcnv  15814  fsumshftm  15822  fsum0diag2  15824  fprod  15985  fprodcnv  16027  bpolyval  16093  ruclem1  16277  odfval  19593  odval  19595  psrass1lem  22043  selvval  22231  mamufval  22510  pm2mpval  22913  isibl  25885  dfitg  25889  dvfsumlem2  26147  fsumdvdsmul  27317  precsexlem3  28360  disjxpin  32843  gsummulsubdishift2s  33304  nmulprop  36553  poimirlem1  38132  poimirlem5  38136  poimirlem15  38146  poimirlem16  38147  poimirlem17  38148  poimirlem19  38150  poimirlem20  38151  poimirlem22  38153  poimirlem24  38155  poimirlem28  38159  evlselv  43183  fphpd  43405  monotuz  43530  oddcomabszz  43533  fnwe2val  43638  fnwe2lem1  43639  dfswapf2  49890  dfinito4  50130