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Theorem sbcel12g 3096
Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcel12g  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )

Proof of Theorem sbcel12g
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2994 . . 3  |-  ( z  =  A  ->  ( [ z  /  x ] B  e.  C  <->  [. A  /  x ]. B  e.  C )
)
2 dfsbcq2 2994 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] y  e.  B  <->  [. A  /  x ]. y  e.  B )
)
32abbidv 2397 . . . 4  |-  ( z  =  A  ->  { y  |  [ z  /  x ] y  e.  B }  =  { y  |  [. A  /  x ]. y  e.  B } )
4 dfsbcq2 2994 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] y  e.  C  <->  [. A  /  x ]. y  e.  C )
)
54abbidv 2397 . . . 4  |-  ( z  =  A  ->  { y  |  [ z  /  x ] y  e.  C }  =  { y  |  [. A  /  x ]. y  e.  C } )
63, 5eleq12d 2351 . . 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 2045 . . . . . 6  |-  F/ x [ z  /  x ] y  e.  B
87nfab 2423 . . . . 5  |-  F/_ x { y  |  [
z  /  x ]
y  e.  B }
9 nfs1v 2045 . . . . . 6  |-  F/ x [ z  /  x ] y  e.  C
109nfab 2423 . . . . 5  |-  F/_ x { y  |  [
z  /  x ]
y  e.  C }
118, 10nfel 2427 . . . 4  |-  F/ x { y  |  [
z  /  x ]
y  e.  B }  e.  { y  |  [
z  /  x ]
y  e.  C }
12 sbab 2405 . . . . 5  |-  ( x  =  z  ->  B  =  { y  |  [
z  /  x ]
y  e.  B }
)
13 sbab 2405 . . . . 5  |-  ( x  =  z  ->  C  =  { y  |  [
z  /  x ]
y  e.  C }
)
1412, 13eleq12d 2351 . . . 4  |-  ( x  =  z  ->  ( B  e.  C  <->  { y  |  [ z  /  x ] y  e.  B }  e.  { y  |  [ z  /  x ] y  e.  C } ) )
1511, 14sbie 1978 . . 3  |-  ( [ z  /  x ] B  e.  C  <->  { y  |  [ z  /  x ] y  e.  B }  e.  { y  |  [ z  /  x ] y  e.  C } )
161, 6, 15vtoclbg 2844 . 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 3082 . . 3  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
18 df-csb 3082 . . 3  |-  [_ A  /  x ]_ C  =  { y  |  [. A  /  x ]. y  e.  C }
1917, 18eleq12i 2348 . 2  |-  ( [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C  <->  { y  |  [. A  /  x ]. y  e.  B }  e.  {
y  |  [. A  /  x ]. y  e.  C } )
2016, 19syl6bbr 254 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623   [wsb 1629    e. wcel 1684   {cab 2269   [.wsbc 2991   [_csb 3081
This theorem is referenced by:  sbcnel12g  3098  sbcel1g  3100  sbcel2g  3102  sbccsb2g  3110  fmptdF  23221  csbxpgVD  28670  csbrngVD  28672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-csb 3082
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