Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbie Structured version   Visualization version   GIF version

Theorem csbie 3918
 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 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
csbie 𝐴 / 𝑥𝐵 = 𝐶
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem csbie
StepHypRef Expression
1 csbie.1 . 2 𝐴 ∈ V
2 nfcv 2977 . 2 𝑥𝐶
3 csbie.2 . 2 (𝑥 = 𝐴𝐵 = 𝐶)
41, 2, 3csbief 3917 1 𝐴 / 𝑥𝐵 = 𝐶
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  Vcvv 3495  ⦋csb 3883 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-sbc 3773  df-csb 3884 This theorem is referenced by:  pofun  5486  eqerlem  8317  mptnn0fsuppd  13360  fsum  15071  fsumcnv  15122  fsumshftm  15130  fsum0diag2  15132  fprod  15289  fprodcnv  15331  bpolyval  15397  ruclem1  15578  odfval  18654  odval  18656  psrass1lem  20151  selvval  20325  mamufval  20990  pm2mpval  21397  isibl  24360  dfitg  24364  dvfsumlem2  24618  fsumdvdsmul  25766  disjxpin  30332  poimirlem1  34887  poimirlem5  34891  poimirlem15  34901  poimirlem16  34902  poimirlem17  34903  poimirlem19  34905  poimirlem20  34906  poimirlem22  34908  poimirlem24  34910  poimirlem28  34914  fphpd  39406  monotuz  39531  oddcomabszz  39534  fnwe2val  39642  fnwe2lem1  39643
 Copyright terms: Public domain W3C validator