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Theorem csbid 3391
Description: Analog of sbid 1949 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 3384 . 2  |-  [_ x  /  x ]_ A  =  { y  |  [. x  /  x ]. y  e.  A }
2 sbcid 3298 . . 3  |-  ( [. x  /  x ]. y  e.  A  <->  y  e.  A
)
32abbii 2583 . 2  |-  { y  |  [. x  /  x ]. y  e.  A }  =  { y  |  y  e.  A }
4 abid2 2589 . 2  |-  { y  |  y  e.  A }  =  A
51, 3, 43eqtri 2483 1  |-  [_ x  /  x ]_ A  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   {cab 2436   [.wsbc 3281   [_csb 3383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-sbc 3282  df-csb 3384
This theorem is referenced by:  csbeq1a  3392  fvmpt2i  5876  fvmpt2curryd  6887  disji2f  26052  disjif2  26056  disjabrex  26057  disjabrexf  26058  fvmpt2f  26106  gsummpt2co  26380  measiuns  26762  fphpd  29290  gsummoncoe1  30982  gsumply1eq  30991
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