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Theorem sbcel12 3740
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 3238 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] B  e.  C  <->  [. A  /  x ]. B  e.  C )
)
2 dfsbcq2 3238 . . . . . 6  |-  ( z  =  A  ->  ( [ z  /  x ] y  e.  B  <->  [. A  /  x ]. y  e.  B )
)
32abbidv 2570 . . . . 5  |-  ( z  =  A  ->  { y  |  [ z  /  x ] y  e.  B }  =  { y  |  [. A  /  x ]. y  e.  B } )
4 dfsbcq2 3238 . . . . . 6  |-  ( z  =  A  ->  ( [ z  /  x ] y  e.  C  <->  [. A  /  x ]. y  e.  C )
)
54abbidv 2570 . . . . 5  |-  ( z  =  A  ->  { y  |  [ z  /  x ] y  e.  C }  =  { y  |  [. A  /  x ]. y  e.  C } )
63, 5eleq12d 2524 . . . 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 2267 . . . . . . 7  |-  F/ x [ z  /  x ] y  e.  B
87nfab 2597 . . . . . 6  |-  F/_ x { y  |  [
z  /  x ]
y  e.  B }
9 nfs1v 2267 . . . . . . 7  |-  F/ x [ z  /  x ] y  e.  C
109nfab 2597 . . . . . 6  |-  F/_ x { y  |  [
z  /  x ]
y  e.  C }
118, 10nfel 2605 . . . . 5  |-  F/ x { y  |  [
z  /  x ]
y  e.  B }  e.  { y  |  [
z  /  x ]
y  e.  C }
12 sbab 2579 . . . . . 6  |-  ( x  =  z  ->  B  =  { y  |  [
z  /  x ]
y  e.  B }
)
13 sbab 2579 . . . . . 6  |-  ( x  =  z  ->  C  =  { y  |  [
z  /  x ]
y  e.  C }
)
1412, 13eleq12d 2524 . . . . 5  |-  ( x  =  z  ->  ( B  e.  C  <->  { y  |  [ z  /  x ] y  e.  B }  e.  { y  |  [ z  /  x ] y  e.  C } ) )
1511, 14sbie 2238 . . . 4  |-  ( [ z  /  x ] B  e.  C  <->  { y  |  [ z  /  x ] y  e.  B }  e.  { y  |  [ z  /  x ] y  e.  C } )
161, 6, 15vtoclbg 3076 . . 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 3332 . . . 4  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
18 df-csb 3332 . . . 4  |-  [_ A  /  x ]_ C  =  { y  |  [. A  /  x ]. y  e.  C }
1917, 18eleq12i 2523 . . 3  |-  ( [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C  <->  { y  |  [. A  /  x ]. y  e.  B }  e.  {
y  |  [. A  /  x ]. y  e.  C } )
2016, 19syl6bbr 271 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
21 sbcex 3245 . . . 4  |-  ( [. A  /  x ]. B  e.  C  ->  A  e. 
_V )
2221con3i 142 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. B  e.  C
)
23 noel 3703 . . . 4  |-  -.  [_ A  /  x ]_ B  e.  (/)
24 csbprc 3738 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ C  =  (/) )
2524eleq2d 2515 . . . 4  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ B  e.  [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B  e.  (/) ) )
2623, 25mtbiri 309 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[_ A  /  x ]_ B  e.  [_ A  /  x ]_ C )
2722, 262falsed 357 . 2  |-  ( -.  A  e.  _V  ->  (
[. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
2820, 27pm2.61i 169 1  |-  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189    = wceq 1448   [wsb 1801    e. wcel 1891   {cab 2438   _Vcvv 3013   [.wsbc 3235   [_csb 3331   (/)c0 3699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1451  df-fal 1454  df-ex 1668  df-nf 1672  df-sb 1802  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-in 3379  df-ss 3386  df-nul 3700
This theorem is referenced by:  sbcnel12g  3742  sbcel1g  3744  sbcel2  3746  sbccsb2  3762  csbmpt12  4708  ixpsnval  7512  fmptdF  28264  csbmpt22g  31734  csbfinxpg  31782  finixpnum  31932
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