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Theorem sbcel1g 3805
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 3801 . 2  |-  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C )
2 csbconstg 3409 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ C  =  C )
32eleq2d 2493 . 2  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B  e.  C
) )
41, 3syl5bb 261 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 188    e. wcel 1869   [.wsbc 3300   [_csb 3396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3763
This theorem is referenced by:  rspcsbela  3824  fprodcllemf  14005  wunnat  15854  catcfuccl  15997  nbgraopALT  25144  rusgrasn  25665  esumpfinvalf  28899  esum2dlem  28915  measiuns  29041  bj-sbel1  31470  finixpnum  31850  renegclALT  32460  cdlemk35s  34429  ellimcabssub0  37523
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