Theorem csbie 3559

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 2764	. 2 ⊢ Ⅎ𝑥𝐶
3		csbie.2	. 2 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
4	1, 2, 3	csbief 3558	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ⦋csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534

This theorem is referenced by:  pofun  5051  eqerlem  7776  mptnn0fsuppd  12798  fsum  14451  fsumcnv  14504  fsumshftm  14513  fsum0diag2  14515  fprod  14671  fprodcnv  14713  bpolyval  14780  ruclem1  14960  odval  17953  psrass1lem  19377  mamufval  20191  pm2mpval  20600  isibl  23532  dfitg  23536  dvfsumlem2  23790  fsumdvdsmul  24921  disjxpin  29401  poimirlem1  33410  poimirlem5  33414  poimirlem15  33424  poimirlem16  33425  poimirlem17  33426  poimirlem19  33428  poimirlem20  33429  poimirlem22  33431  poimirlem24  33433  poimirlem28  33437  fphpd  37380  monotuz  37506  oddcomabszz  37509  fnwe2val  37619  fnwe2lem1  37620