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Theorem sbcel2 3836
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 3828 . . 3  |-  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C )
2 csbconstg 3453 . . . 4  |-  ( A  e.  _V  ->  [_ A  /  x ]_ B  =  B )
32eleq1d 2536 . . 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 3346 . . . 4  |-  ( [. A  /  x ]. B  e.  C  ->  A  e. 
_V )
65con3i 135 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. B  e.  C
)
7 noel 3794 . . . 4  |-  -.  B  e.  (/)
8 csbprc 3826 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ C  =  (/) )
98eleq2d 2537 . . . 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 1767   _Vcvv 3118   [.wsbc 3336   [_csb 3440   (/)c0 3790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-in 3488  df-ss 3495  df-nul 3791
This theorem is referenced by:  csbcom  3842  sbccsb  3854  sbnfc2  3859  csbab  3860  sbcssg  3944  csbuni  4279  csbxp  5087  csbdm  5203  issubc  15082  nbgraopALT  24247  csbcom2fi  30462  bj-sbeq  33950  bj-sbceqgALT  33951  bj-sels  34002
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