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Theorem csbid 3080
Description: Analog of sbid 1785 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 3073 . 2  |-  [_ x  /  x ]_ A  =  { y  |  [. x  /  x ]. y  e.  A }
2 sbcid 2993 . . 3  |-  ( [. x  /  x ]. y  e.  A  <->  y  e.  A
)
32abbii 2305 . 2  |-  { y  |  [. x  /  x ]. y  e.  A }  =  { y  |  y  e.  A }
4 abid2 2310 . 2  |-  { y  |  y  e.  A }  =  A
51, 3, 43eqtri 2214 1  |-  [_ x  /  x ]_ A  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1364    e. wcel 2160   {cab 2175   [.wsbc 2977   [_csb 3072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-sbc 2978  df-csb 3073
This theorem is referenced by:  csbeq1a  3081  fvmpt2  5620  fsumsplitf  11448  ctiunctlemfo  12490
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