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Theorem sbcel2gOLD 36761
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 3806 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 36760 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
2 csbconstg 3408 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  B )
32eleq1d 2491 . 2  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C  <->  B  e.  [_ A  /  x ]_ C ) )
41, 3bitrd 256 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 187    e. wcel 1868   [.wsbc 3299   [_csb 3395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-sbc 3300  df-csb 3396
This theorem is referenced by:  sbcssOLD  36762  csbabgOLD  37069  csbunigOLD  37070  csbxpgOLD  37072  csbrngOLD  37075  sbcssgVD  37138  csbingVD  37139  csbunigVD  37153
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