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Theorem csbid 2916
Description: Analog of sbid 1698 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 2910 . 2 𝑥 / 𝑥𝐴 = {𝑦[𝑥 / 𝑥]𝑦𝐴}
2 sbcid 2831 . . 3 ([𝑥 / 𝑥]𝑦𝐴𝑦𝐴)
32abbii 2195 . 2 {𝑦[𝑥 / 𝑥]𝑦𝐴} = {𝑦𝑦𝐴}
4 abid2 2200 . 2 {𝑦𝑦𝐴} = 𝐴
51, 3, 43eqtri 2106 1 𝑥 / 𝑥𝐴 = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1285  wcel 1434  {cab 2068  [wsbc 2816  csb 2909
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-sbc 2817  df-csb 2910
This theorem is referenced by:  csbeq1a  2917  fvmpt2  5286
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