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Theorem csbeq1 2849
 Description: Analog of dfsbcq 2760 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbeq1 (A = BA / x𝐶 = B / x𝐶)

Proof of Theorem csbeq1
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 2760 . . 3 (A = B → ([A / x]y 𝐶[B / x]y 𝐶))
21abbidv 2152 . 2 (A = B → {y[A / x]y 𝐶} = {y[B / x]y 𝐶})
3 df-csb 2847 . 2 A / x𝐶 = {y[A / x]y 𝐶}
4 df-csb 2847 . 2 B / x𝐶 = {y[B / x]y 𝐶}
52, 3, 43eqtr4g 2094 1 (A = BA / x𝐶 = B / x𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  {cab 2023  [wsbc 2758  ⦋csb 2846 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-sbc 2759  df-csb 2847 This theorem is referenced by:  csbeq1d  2852  csbeq1a  2854  csbiebg  2883  sbcnestgf  2891  cbvralcsf  2902  cbvrexcsf  2903  cbvreucsf  2904  cbvrabcsf  2905  csbing  3138  sbcbrg  3804  csbopabg  3826  pofun  4040  csbima12g  4629  csbiotag  4838  fvmpts  5193  fvmpt2  5197  mptfvex  5199  fmptcof  5274  fmptcos  5275  fliftfuns  5381  csbriotag  5423  csbov123g  5485  eqerlem  6073  qliftfuns  6126
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