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Theorem sbcel2 3794
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 3786 . . 3  |-  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C )
2 csbconstg 3411 . . . 4  |-  ( A  e.  _V  ->  [_ A  /  x ]_ B  =  B )
32eleq1d 2523 . . 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 3304 . . . 4  |-  ( [. A  /  x ]. B  e.  C  ->  A  e. 
_V )
65con3i 135 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. B  e.  C
)
7 noel 3752 . . . 4  |-  -.  B  e.  (/)
8 csbprc 3784 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ C  =  (/) )
98eleq2d 2524 . . . 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 1758   _Vcvv 3078   [.wsbc 3294   [_csb 3398   (/)c0 3748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-in 3446  df-ss 3453  df-nul 3749
This theorem is referenced by:  csbcom  3800  sbccsb  3812  sbnfc2  3817  csbab  3818  sbcssg  3901  csbuni  4230  csbxp  5029  csbdm  5145  issubc  14871  csbcom2fi  29109  bj-sbeq  32760  bj-sbceqgALT  32761  bj-sels  32812
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