MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcel12g Structured version   Unicode version

Theorem sbcel12g 3266
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 3164 . . 3  |-  ( z  =  A  ->  ( [ z  /  x ] B  e.  C  <->  [. A  /  x ]. B  e.  C )
)
2 dfsbcq2 3164 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] y  e.  B  <->  [. A  /  x ]. y  e.  B )
)
32abbidv 2550 . . . 4  |-  ( z  =  A  ->  { y  |  [ z  /  x ] y  e.  B }  =  { y  |  [. A  /  x ]. y  e.  B } )
4 dfsbcq2 3164 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] y  e.  C  <->  [. A  /  x ]. y  e.  C )
)
54abbidv 2550 . . . 4  |-  ( z  =  A  ->  { y  |  [ z  /  x ] y  e.  C }  =  { y  |  [. A  /  x ]. y  e.  C } )
63, 5eleq12d 2504 . . 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 2182 . . . . . 6  |-  F/ x [ z  /  x ] y  e.  B
87nfab 2576 . . . . 5  |-  F/_ x { y  |  [
z  /  x ]
y  e.  B }
9 nfs1v 2182 . . . . . 6  |-  F/ x [ z  /  x ] y  e.  C
109nfab 2576 . . . . 5  |-  F/_ x { y  |  [
z  /  x ]
y  e.  C }
118, 10nfel 2580 . . . 4  |-  F/ x { y  |  [
z  /  x ]
y  e.  B }  e.  { y  |  [
z  /  x ]
y  e.  C }
12 sbab 2558 . . . . 5  |-  ( x  =  z  ->  B  =  { y  |  [
z  /  x ]
y  e.  B }
)
13 sbab 2558 . . . . 5  |-  ( x  =  z  ->  C  =  { y  |  [
z  /  x ]
y  e.  C }
)
1412, 13eleq12d 2504 . . . 4  |-  ( x  =  z  ->  ( B  e.  C  <->  { y  |  [ z  /  x ] y  e.  B }  e.  { y  |  [ z  /  x ] y  e.  C } ) )
1511, 14sbie 2149 . . 3  |-  ( [ z  /  x ] B  e.  C  <->  { y  |  [ z  /  x ] y  e.  B }  e.  { y  |  [ z  /  x ] y  e.  C } )
161, 6, 15vtoclbg 3012 . 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 3252 . . 3  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
18 df-csb 3252 . . 3  |-  [_ A  /  x ]_ C  =  { y  |  [. A  /  x ]. y  e.  C }
1917, 18eleq12i 2501 . 2  |-  ( [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C  <->  { y  |  [. A  /  x ]. y  e.  B }  e.  {
y  |  [. A  /  x ]. y  e.  C } )
2016, 19syl6bbr 255 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 177    = wceq 1652   [wsb 1658    e. wcel 1725   {cab 2422   [.wsbc 3161   [_csb 3251
This theorem is referenced by:  sbcnel12g  3268  sbcel1g  3270  sbcel2g  3272  sbccsb2g  3280  fmptdF  24069  csbxpgVD  29006  csbrngVD  29008
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-sbc 3162  df-csb 3252
  Copyright terms: Public domain W3C validator