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Theorem sbcel1g 3826
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 sbcel12 3822 . 2  |-  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C )
2 csbconstg 3433 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ C  =  C )
32eleq2d 2524 . 2  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B  e.  C
) )
41, 3syl5bb 257 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  C )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1823   [.wsbc 3324   [_csb 3420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784
This theorem is referenced by:  rspcsbela  3845  wunnat  15444  catcfuccl  15587  nbgraopALT  24626  rusgrasn  25147  esumpfinvalf  28305  esum2dlem  28321  measiuns  28425  finixpnum  30278  fprodcllemf  31830  ellimcabssub0  31862  bj-sbel1  34873  renegclALT  35091  cdlemk35s  37060
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