Theorem csbie 3754

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019.)

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

Step	Hyp	Ref	Expression
1		csbie.1	. 2 ⊢ 𝐴 ∈ V
2		nfcv 2941	. 2 ⊢ Ⅎ𝑥𝐶
3		csbie.2	. 2 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
4	1, 2, 3	csbief 3753	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶

Syntax hints:   → wi 4   = wceq 1653   ∈ wcel 2157  Vcvv 3385  ⦋csb 3728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-sbc 3634  df-csb 3729

This theorem is referenced by:  pofun  5249  eqerlem  8016  mptnn0fsuppd  13052  fsum  14792  fsumcnv  14843  fsumshftm  14851  fsum0diag2  14853  fprod  15008  fprodcnv  15050  bpolyval  15116  ruclem1  15296  odval  18266  psrass1lem  19700  mamufval  20516  pm2mpval  20928  isibl  23873  dfitg  23877  dvfsumlem2  24131  fsumdvdsmul  25273  disjxpin  29918  poimirlem1  33899  poimirlem5  33903  poimirlem15  33913  poimirlem16  33914  poimirlem17  33915  poimirlem19  33917  poimirlem20  33918  poimirlem22  33920  poimirlem24  33922  poimirlem28  33926  fphpd  38166  monotuz  38291  oddcomabszz  38294  fnwe2val  38404  fnwe2lem1  38405