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Theorem csbid 2924
Description: Analog of sbid 1699 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbid  |-  [_ x  /  x ]_ A  =  A

Proof of Theorem csbid
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-csb 2918 . 2  |-  [_ x  /  x ]_ A  =  { y  |  [. x  /  x ]. y  e.  A }
2 sbcid 2839 . . 3  |-  ( [. x  /  x ]. y  e.  A  <->  y  e.  A
)
32abbii 2198 . 2  |-  { y  |  [. x  /  x ]. y  e.  A }  =  { y  |  y  e.  A }
4 abid2 2203 . 2  |-  { y  |  y  e.  A }  =  A
51, 3, 43eqtri 2107 1  |-  [_ x  /  x ]_ A  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434   {cab 2069   [.wsbc 2824   [_csb 2917
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 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-sbc 2825  df-csb 2918
This theorem is referenced by:  csbeq1a  2925  fvmpt2  5306
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