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Theorem sbcel2gOLD 36349
Description: Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) Obsolete as of 18-Aug-2018. Use sbcel2 3785 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbcel2gOLD  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  B  e.  [_ A  /  x ]_ C ) )
Distinct variable group:    x, B
Allowed substitution hints:    A( x)    C( x)    V( x)

Proof of Theorem sbcel2gOLD
StepHypRef Expression
1 sbcel12gOLD 36348 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
2 csbconstg 3388 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  B )
32eleq1d 2473 . 2  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C  <->  B  e.  [_ A  /  x ]_ C ) )
41, 3bitrd 255 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  B  e.  [_ A  /  x ]_ C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    e. wcel 1844   [.wsbc 3279   [_csb 3375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-v 3063  df-sbc 3280  df-csb 3376
This theorem is referenced by:  sbcssOLD  36350  csbabgOLD  36658  csbunigOLD  36659  csbxpgOLD  36661  csbrngOLD  36664  sbcssgVD  36727  csbingVD  36728  csbunigVD  36742
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