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Theorem sbcel12 3786
Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbcel12  |-  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C )

Proof of Theorem sbcel12
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3297 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] B  e.  C  <->  [. A  /  x ]. B  e.  C )
)
2 dfsbcq2 3297 . . . . . 6  |-  ( z  =  A  ->  ( [ z  /  x ] y  e.  B  <->  [. A  /  x ]. y  e.  B )
)
32abbidv 2590 . . . . 5  |-  ( z  =  A  ->  { y  |  [ z  /  x ] y  e.  B }  =  { y  |  [. A  /  x ]. y  e.  B } )
4 dfsbcq2 3297 . . . . . 6  |-  ( z  =  A  ->  ( [ z  /  x ] y  e.  C  <->  [. A  /  x ]. y  e.  C )
)
54abbidv 2590 . . . . 5  |-  ( z  =  A  ->  { y  |  [ z  /  x ] y  e.  C }  =  { y  |  [. A  /  x ]. y  e.  C } )
63, 5eleq12d 2536 . . . 4  |-  ( z  =  A  ->  ( { y  |  [
z  /  x ]
y  e.  B }  e.  { y  |  [
z  /  x ]
y  e.  C }  <->  { y  |  [. A  /  x ]. y  e.  B }  e.  {
y  |  [. A  /  x ]. y  e.  C } ) )
7 nfs1v 2151 . . . . . . 7  |-  F/ x [ z  /  x ] y  e.  B
87nfab 2620 . . . . . 6  |-  F/_ x { y  |  [
z  /  x ]
y  e.  B }
9 nfs1v 2151 . . . . . . 7  |-  F/ x [ z  /  x ] y  e.  C
109nfab 2620 . . . . . 6  |-  F/_ x { y  |  [
z  /  x ]
y  e.  C }
118, 10nfel 2629 . . . . 5  |-  F/ x { y  |  [
z  /  x ]
y  e.  B }  e.  { y  |  [
z  /  x ]
y  e.  C }
12 sbab 2601 . . . . . 6  |-  ( x  =  z  ->  B  =  { y  |  [
z  /  x ]
y  e.  B }
)
13 sbab 2601 . . . . . 6  |-  ( x  =  z  ->  C  =  { y  |  [
z  /  x ]
y  e.  C }
)
1412, 13eleq12d 2536 . . . . 5  |-  ( x  =  z  ->  ( B  e.  C  <->  { y  |  [ z  /  x ] y  e.  B }  e.  { y  |  [ z  /  x ] y  e.  C } ) )
1511, 14sbie 2110 . . . 4  |-  ( [ z  /  x ] B  e.  C  <->  { y  |  [ z  /  x ] y  e.  B }  e.  { y  |  [ z  /  x ] y  e.  C } )
161, 6, 15vtoclbg 3137 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B  e.  C  <->  { y  |  [. A  /  x ]. y  e.  B }  e.  {
y  |  [. A  /  x ]. y  e.  C } ) )
17 df-csb 3399 . . . 4  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
18 df-csb 3399 . . . 4  |-  [_ A  /  x ]_ C  =  { y  |  [. A  /  x ]. y  e.  C }
1917, 18eleq12i 2533 . . 3  |-  ( [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C  <->  { y  |  [. A  /  x ]. y  e.  B }  e.  {
y  |  [. A  /  x ]. y  e.  C } )
2016, 19syl6bbr 263 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
21 sbcex 3304 . . . 4  |-  ( [. A  /  x ]. B  e.  C  ->  A  e. 
_V )
2221con3i 135 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. B  e.  C
)
23 noel 3752 . . . 4  |-  -.  [_ A  /  x ]_ B  e.  (/)
24 csbprc 3784 . . . . . 6  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ C  =  (/) )
2524eleq2d 2524 . . . . 5  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ B  e.  [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B  e.  (/) ) )
2625notbid 294 . . . 4  |-  ( -.  A  e.  _V  ->  ( -.  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C  <->  -. 
[_ A  /  x ]_ B  e.  (/) ) )
2723, 26mpbiri 233 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[_ A  /  x ]_ B  e.  [_ A  /  x ]_ C )
2822, 272falsed 351 . 2  |-  ( -.  A  e.  _V  ->  (
[. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
2920, 28pm2.61i 164 1  |-  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1370   [wsb 1702    e. wcel 1758   {cab 2439   _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:  sbcnel12g  3789  sbcel1g  3792  sbcel2  3794  sbccsb2  3814  csbmpt12  4733  ixpsnval  7379  fmptdF  26150  finixpnum  28585
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