Theorem csbie 3885

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 3851	. 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 2848	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
5	2, 4	sbcie 3783	. . 3 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)
6	5	abbii 2829	. 2 ⊢ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ 𝑦 ∈ 𝐶}
7		abid2 2899	. 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐶} = 𝐶
8	1, 6, 7	3eqtri 2789	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {cab 2740  Vcvv 3453  [wsbc 3742  ⦋csb 3850

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-sbc 3743  df-csb 3851

This theorem is used by:  pofun  5585  eqerlem  8735  mptnn0fsuppd  14064  fsum  15808  fsumcnv  15861  fsumshftm  15869  fsum0diag2  15871  fprod  16032  fprodcnv  16074  bpolyval  16139  ruclem1  16323  odfval  19663  odval  19665  psrass1lem  22152  selvval  22340  mamufval  22618  pm2mpval  23024  isibl  25997  dfitg  26001  dvfsumlem2  26259  fsumdvdsmul  27432  precsexlem3  28475  disjxpin  33063  gsummulsubdishift2s  33513  nmulprop  36772  poimirlem1  38372  poimirlem5  38376  poimirlem15  38386  poimirlem16  38387  poimirlem17  38388  poimirlem19  38390  poimirlem20  38391  poimirlem22  38393  poimirlem24  38395  poimirlem28  38399  evlselv  43437  fphpd  43659  monotuz  43784  oddcomabszz  43787  fnwe2val  43892  fnwe2lem1  43893  dfswapf2  50189  dfinito4  50429