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Theorem csbid 3274
Description: Analog of sbid 1950 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 3268 . 2  |-  [_ x  /  x ]_ A  =  { y  |  [. x  /  x ]. y  e.  A }
2 sbcid 3183 . . 3  |-  ( [. x  /  x ]. y  e.  A  <->  y  e.  A
)
32abbii 2554 . 2  |-  { y  |  [. x  /  x ]. y  e.  A }  =  { y  |  y  e.  A }
4 abid2 2559 . 2  |-  { y  |  y  e.  A }  =  A
51, 3, 43eqtri 2466 1  |-  [_ x  /  x ]_ A  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1727   {cab 2428   [.wsbc 3167   [_csb 3267
This theorem is referenced by:  csbeq1a  3275  fvmpt2i  5840  disji2f  24050  disjif2  24054  disjabrex  24055  disjabrexf  24056  fvmpt2f  24103  measiuns  24602  fphpd  26915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-sbc 3168  df-csb 3268
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