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Theorem csbid 3016
 Description: Analog of sbid 1748 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 3009 . 2 𝑥 / 𝑥𝐴 = {𝑦[𝑥 / 𝑥]𝑦𝐴}
2 sbcid 2929 . . 3 ([𝑥 / 𝑥]𝑦𝐴𝑦𝐴)
32abbii 2256 . 2 {𝑦[𝑥 / 𝑥]𝑦𝐴} = {𝑦𝑦𝐴}
4 abid2 2261 . 2 {𝑦𝑦𝐴} = 𝐴
51, 3, 43eqtri 2165 1 𝑥 / 𝑥𝐴 = 𝐴
 Colors of variables: wff set class Syntax hints:   = wceq 1332   ∈ wcel 1481  {cab 2126  [wsbc 2914  ⦋csb 3008 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-sbc 2915  df-csb 3009 This theorem is referenced by:  csbeq1a  3017  fvmpt2  5514  fsumsplitf  11232  ctiunctlemfo  12011
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