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Theorem sbcel1g 3270
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 sbcel12g 3266 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
2 csbconstg 3265 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ C  =  C )
32eleq2d 2503 . 2  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B  e.  C
) )
41, 3bitrd 245 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1725   [.wsbc 3161   [_csb 3251
This theorem is referenced by:  rspcsbela  3308  wunnat  14153  catcfuccl  14264  esumpfinvalf  24466  measiuns  24571  cdlemk35s  31734
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-sbc 3162  df-csb 3252
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