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Theorem sbcel2 3839
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 3832 . . 3  |-  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C )
2 csbconstg 3443 . . . 4  |-  ( A  e.  _V  ->  [_ A  /  x ]_ B  =  B )
32eleq1d 2526 . . 3  |-  ( A  e.  _V  ->  ( [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C  <->  B  e.  [_ A  /  x ]_ C ) )
41, 3syl5bb 257 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B  e.  C  <->  B  e.  [_ A  /  x ]_ C ) )
5 sbcex 3337 . . . 4  |-  ( [. A  /  x ]. B  e.  C  ->  A  e. 
_V )
65con3i 135 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. B  e.  C
)
7 noel 3797 . . . 4  |-  -.  B  e.  (/)
8 csbprc 3830 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ C  =  (/) )
98eleq2d 2527 . . . 4  |-  ( -.  A  e.  _V  ->  ( B  e.  [_ A  /  x ]_ C  <->  B  e.  (/) ) )
107, 9mtbiri 303 . . 3  |-  ( -.  A  e.  _V  ->  -.  B  e.  [_ A  /  x ]_ C )
116, 102falsed 351 . 2  |-  ( -.  A  e.  _V  ->  (
[. A  /  x ]. B  e.  C  <->  B  e.  [_ A  /  x ]_ C ) )
124, 11pm2.61i 164 1  |-  ( [. A  /  x ]. B  e.  C  <->  B  e.  [_ A  /  x ]_ C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    e. wcel 1819   _Vcvv 3109   [.wsbc 3327   [_csb 3430   (/)c0 3793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3794
This theorem is referenced by:  csbcom  3845  sbccsb  3856  sbnfc2  3859  csbab  3860  sbcssg  3943  csbuni  4279  csbxp  5090  csbdm  5207  issubc  15251  nbgraopALT  24551  esum2dlem  28264  csbcom2fi  30718  bj-sbeq  34590  bj-sbceqgALT  34591  bj-sels  34642
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