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Theorem sbcel2g 2928
Description: Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.)
Assertion
Ref Expression
sbcel2g  |-  ( 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 sbcel2g
StepHypRef Expression
1 sbcel12g 2922 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
2 csbconstg 2921 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  B )
32eleq1d 2148 . 2  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C  <->  B  e.  [_ A  /  x ]_ C ) )
41, 3bitrd 186 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  B  e.  [_ A  /  x ]_ C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    e. wcel 1434   [.wsbc 2816   [_csb 2909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817  df-csb 2910
This theorem is referenced by:  csbcomg  2930  sbccsbg  2935  sbnfc2  2963  csbabg  2964  sbcssg  3358  csbunig  3617  csbxpg  4447  csbdmg  4557  csbrng  4812  bj-sels  10863
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