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Theorem csbid 3443
Description: Analog of sbid 1965 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 3436 . 2  |-  [_ x  /  x ]_ A  =  { y  |  [. x  /  x ]. y  e.  A }
2 sbcid 3348 . . 3  |-  ( [. x  /  x ]. y  e.  A  <->  y  e.  A
)
32abbii 2601 . 2  |-  { y  |  [. x  /  x ]. y  e.  A }  =  { y  |  y  e.  A }
4 abid2 2607 . 2  |-  { y  |  y  e.  A }  =  A
51, 3, 43eqtri 2500 1  |-  [_ x  /  x ]_ A  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   {cab 2452   [.wsbc 3331   [_csb 3435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-sbc 3332  df-csb 3436
This theorem is referenced by:  csbeq1a  3444  fvmpt2i  5954  fvmpt2curryd  6997  gsummoncoe1  18114  gsumply1eq  18115  disji2f  27108  disjif2  27112  disjabrex  27113  disjabrexf  27114  fvmpt2f  27167  gsummpt2co  27431  measiuns  27825  fphpd  30352
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