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Theorem sbcel12gOLD 36905
Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcel12 3772 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbcel12gOLD  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )

Proof of Theorem sbcel12gOLD
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3270 . . 3  |-  ( z  =  A  ->  ( [ z  /  x ] B  e.  C  <->  [. A  /  x ]. B  e.  C )
)
2 dfsbcq2 3270 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] y  e.  B  <->  [. A  /  x ]. y  e.  B )
)
32abbidv 2569 . . . 4  |-  ( z  =  A  ->  { y  |  [ z  /  x ] y  e.  B }  =  { y  |  [. A  /  x ]. y  e.  B } )
4 dfsbcq2 3270 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] y  e.  C  <->  [. A  /  x ]. y  e.  C )
)
54abbidv 2569 . . . 4  |-  ( z  =  A  ->  { y  |  [ z  /  x ] y  e.  C }  =  { y  |  [. A  /  x ]. y  e.  C } )
63, 5eleq12d 2523 . . 3  |-  ( 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 2266 . . . . . 6  |-  F/ x [ z  /  x ] y  e.  B
87nfab 2596 . . . . 5  |-  F/_ x { y  |  [
z  /  x ]
y  e.  B }
9 nfs1v 2266 . . . . . 6  |-  F/ x [ z  /  x ] y  e.  C
109nfab 2596 . . . . 5  |-  F/_ x { y  |  [
z  /  x ]
y  e.  C }
118, 10nfel 2604 . . . 4  |-  F/ x { y  |  [
z  /  x ]
y  e.  B }  e.  { y  |  [
z  /  x ]
y  e.  C }
12 sbab 2578 . . . . 5  |-  ( x  =  z  ->  B  =  { y  |  [
z  /  x ]
y  e.  B }
)
13 sbab 2578 . . . . 5  |-  ( x  =  z  ->  C  =  { y  |  [
z  /  x ]
y  e.  C }
)
1412, 13eleq12d 2523 . . . 4  |-  ( x  =  z  ->  ( B  e.  C  <->  { y  |  [ z  /  x ] y  e.  B }  e.  { y  |  [ z  /  x ] y  e.  C } ) )
1511, 14sbie 2237 . . 3  |-  ( [ z  /  x ] B  e.  C  <->  { y  |  [ z  /  x ] y  e.  B }  e.  { y  |  [ z  /  x ] y  e.  C } )
161, 6, 15vtoclbg 3108 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  { y  |  [. A  /  x ]. y  e.  B }  e.  {
y  |  [. A  /  x ]. y  e.  C } ) )
17 df-csb 3364 . . 3  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
18 df-csb 3364 . . 3  |-  [_ A  /  x ]_ C  =  { y  |  [. A  /  x ]. y  e.  C }
1917, 18eleq12i 2522 . 2  |-  ( [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C  <->  { y  |  [. A  /  x ]. y  e.  B }  e.  {
y  |  [. A  /  x ]. y  e.  C } )
2016, 19syl6bbr 267 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    = wceq 1444   [wsb 1797    e. wcel 1887   {cab 2437   [.wsbc 3267   [_csb 3363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-sbc 3268  df-csb 3364
This theorem is referenced by:  sbcel2gOLD  36906  csbxpgVD  37291  csbrngVD  37293
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