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Theorem sbcel1g 3787
Description: Move proper substitution in and out of a membership relation. Note that the scope of  [. A  /  x ]. is the wff  B  e.  C, whereas the scope of  [_ A  /  x ]_ is the class  B. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
sbcel1g  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  C )
)
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)    V( x)

Proof of Theorem sbcel1g
StepHypRef Expression
1 sbcel12 3783 . 2  |-  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C )
2 csbconstg 3387 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ C  =  C )
32eleq2d 2524 . 2  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B  e.  C
) )
41, 3syl5bb 265 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  C )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    e. wcel 1897   [.wsbc 3278   [_csb 3374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-in 3422  df-ss 3429  df-nul 3743
This theorem is referenced by:  rspcsbela  3806  fprodcllemf  14060  wunnat  15909  catcfuccl  16052  nbgraopALT  25200  rusgrasn  25721  esumpfinvalf  28945  esum2dlem  28961  measiuns  29087  bj-sbel1  31551  csbopg2  31769  csbfinxpg  31824  finixpnum  31974  renegclALT  32579  cdlemk35s  34548  ellimcabssub0  37734
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