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Theorem csbid 3222
Description: Analog of sbid 1943 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 3216 . 2  |-  [_ x  /  x ]_ A  =  { y  |  [. x  /  x ]. y  e.  A }
2 sbcid 3141 . . 3  |-  ( [. x  /  x ]. y  e.  A  <->  y  e.  A
)
32abbii 2520 . 2  |-  { y  |  [. x  /  x ]. y  e.  A }  =  { y  |  y  e.  A }
4 abid2 2525 . 2  |-  { y  |  y  e.  A }  =  A
51, 3, 43eqtri 2432 1  |-  [_ x  /  x ]_ A  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   {cab 2394   [.wsbc 3125   [_csb 3215
This theorem is referenced by:  csbeq1a  3223  fvmpt2i  5774  disji2f  23976  disjif2  23980  disjabrex  23981  disjabrexf  23982  fvmpt2f  24029  measiuns  24528  fphpd  26771
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-sbc 3126  df-csb 3216
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