Theorem csbie 3887

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 3853	. 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 2847	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
5	2, 4	sbcie 3784	. . 3 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)
6	5	abbii 2828	. 2 ⊢ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ 𝑦 ∈ 𝐶}
7		abid2 2898	. 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐶} = 𝐶
8	1, 6, 7	3eqtri 2788	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2141  {cab 2739  Vcvv 3453  [wsbc 3743  ⦋csb 3852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-sbc 3744  df-csb 3853

This theorem is referenced by:  pofun  5587  eqerlem  8729  mptnn0fsuppd  14034  fsum  15771  fsumcnv  15824  fsumshftm  15832  fsum0diag2  15834  fprod  15995  fprodcnv  16037  bpolyval  16102  ruclem1  16286  odfval  19601  odval  19603  psrass1lem  22062  selvval  22250  mamufval  22528  pm2mpval  22931  isibl  25903  dfitg  25907  dvfsumlem2  26165  fsumdvdsmul  27335  precsexlem3  28378  disjxpin  32899  gsummulsubdishift2s  33357  nmulprop  36648  poimirlem1  38238  poimirlem5  38242  poimirlem15  38252  poimirlem16  38253  poimirlem17  38254  poimirlem19  38256  poimirlem20  38257  poimirlem22  38259  poimirlem24  38261  poimirlem28  38265  evlselv  43291  fphpd  43513  monotuz  43638  oddcomabszz  43641  fnwe2val  43746  fnwe2lem1  43747  dfswapf2  50006  dfinito4  50246