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Theorem sbcel2 3790
Description: Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbcel2  |-  ( [. A  /  x ]. B  e.  C  <->  B  e.  [_ A  /  x ]_ C )
Distinct variable group:    x, B
Allowed substitution hints:    A( x)    C( x)

Proof of Theorem sbcel2
StepHypRef Expression
1 sbcel12 3784 . . 3  |-  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C )
2 csbconstg 3388 . . . 4  |-  ( A  e.  _V  ->  [_ A  /  x ]_ B  =  B )
32eleq1d 2524 . . 3  |-  ( A  e.  _V  ->  ( [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C  <->  B  e.  [_ A  /  x ]_ C ) )
41, 3syl5bb 265 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B  e.  C  <->  B  e.  [_ A  /  x ]_ C ) )
5 sbcex 3289 . . . 4  |-  ( [. A  /  x ]. B  e.  C  ->  A  e. 
_V )
65con3i 142 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. B  e.  C
)
7 noel 3747 . . . 4  |-  -.  B  e.  (/)
8 csbprc 3782 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ C  =  (/) )
98eleq2d 2525 . . . 4  |-  ( -.  A  e.  _V  ->  ( B  e.  [_ A  /  x ]_ C  <->  B  e.  (/) ) )
107, 9mtbiri 309 . . 3  |-  ( -.  A  e.  _V  ->  -.  B  e.  [_ A  /  x ]_ C )
116, 102falsed 357 . 2  |-  ( -.  A  e.  _V  ->  (
[. A  /  x ]. B  e.  C  <->  B  e.  [_ A  /  x ]_ C ) )
124, 11pm2.61i 169 1  |-  ( [. A  /  x ]. B  e.  C  <->  B  e.  [_ A  /  x ]_ C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189    e. wcel 1898   _Vcvv 3057   [.wsbc 3279   [_csb 3375   (/)c0 3743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-in 3423  df-ss 3430  df-nul 3744
This theorem is referenced by:  csbcom  3795  sbccsb  3805  sbnfc2  3808  csbab  3809  sbcssg  3892  csbuni  4240  csbxp  4935  csbdm  5048  issubc  15789  nbgraopALT  25201  esum2dlem  28962  bj-sbeq  31548  bj-sbceqgALT  31549  bj-sels  31601  f1omptsnlem  31783  csbcom2fi  32414  disjinfi  37506  iccelpart  38785
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