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Theorem csbid 3164
Description: Analog of sbid 1927 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 3158 . 2  |-  [_ x  /  x ]_ A  =  { y  |  [. x  /  x ]. y  e.  A }
2 sbsbc 3071 . . . 4  |-  ( [ x  /  x ]
y  e.  A  <->  [. x  /  x ]. y  e.  A
)
3 sbid 1927 . . . 4  |-  ( [ x  /  x ]
y  e.  A  <->  y  e.  A )
42, 3bitr3i 242 . . 3  |-  ( [. x  /  x ]. y  e.  A  <->  y  e.  A
)
54abbii 2470 . 2  |-  { y  |  [. x  /  x ]. y  e.  A }  =  { y  |  y  e.  A }
6 abid2 2475 . 2  |-  { y  |  y  e.  A }  =  A
71, 5, 63eqtri 2382 1  |-  [_ x  /  x ]_ A  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1642   [wsb 1648    e. wcel 1710   {cab 2344   [.wsbc 3067   [_csb 3157
This theorem is referenced by:  csbeq1a  3165  fvmpt2i  5690  disji2f  23218  disjif2  23222  disjabrex  23223  disjabrexf  23224  fvmpt2f  23275  measiuns  23835  fphpd  26222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-sbc 3068  df-csb 3158
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