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Theorem sbcel2g 3102
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 3096 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
2 csbconstg 3095 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  B )
32eleq1d 2349 . 2  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C  <->  B  e.  [_ A  /  x ]_ C ) )
41, 3bitrd 244 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 176    e. wcel 1684   [.wsbc 2991   [_csb 3081
This theorem is referenced by:  csbcomg  3104  sbccsbg  3109  sbnfc2  3141  csbabg  3142  sbcss  3564  csbunig  3835  csbxpg  4716  csbrng  4923  issubc  13712  csbdmg  27980  sbcssOLD  28306  sbcssVD  28659  csbingVD  28660  csbunigVD  28674
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-csb 3082
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