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Theorem csbid 3088
Description: Analog of sbid 1863 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 3082 . 2  |-  [_ x  /  x ]_ A  =  { y  |  [. x  /  x ]. y  e.  A }
2 sbsbc 2995 . . . 4  |-  ( [ x  /  x ]
y  e.  A  <->  [. x  /  x ]. y  e.  A
)
3 sbid 1863 . . . 4  |-  ( [ x  /  x ]
y  e.  A  <->  y  e.  A )
42, 3bitr3i 242 . . 3  |-  ( [. x  /  x ]. y  e.  A  <->  y  e.  A
)
54abbii 2395 . 2  |-  { y  |  [. x  /  x ]. y  e.  A }  =  { y  |  y  e.  A }
6 abid2 2400 . 2  |-  { y  |  y  e.  A }  =  A
71, 5, 63eqtri 2307 1  |-  [_ x  /  x ]_ A  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1623   [wsb 1629    e. wcel 1684   {cab 2269   [.wsbc 2991   [_csb 3081
This theorem is referenced by:  csbeq1a  3089  fvmpt2i  5607  fvmpt2f  23224  disji2f  23354  disjif2  23358  disjabrex  23359  disjabrexf  23360  measiuns  23544  fphpd  26899
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-sbc 2992  df-csb 3082
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