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Theorem csbid 2938
Description: Analog of sbid 1704 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbid 𝑥 / 𝑥𝐴 = 𝐴

Proof of Theorem csbid
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-csb 2932 . 2 𝑥 / 𝑥𝐴 = {𝑦[𝑥 / 𝑥]𝑦𝐴}
2 sbcid 2853 . . 3 ([𝑥 / 𝑥]𝑦𝐴𝑦𝐴)
32abbii 2203 . 2 {𝑦[𝑥 / 𝑥]𝑦𝐴} = {𝑦𝑦𝐴}
4 abid2 2208 . 2 {𝑦𝑦𝐴} = 𝐴
51, 3, 43eqtri 2112 1 𝑥 / 𝑥𝐴 = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1289  wcel 1438  {cab 2074  [wsbc 2838  csb 2931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-sbc 2839  df-csb 2932
This theorem is referenced by:  csbeq1a  2939  fvmpt2  5370  fsumsplitf  10765
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