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Theorem sbceqg 3097
Description: Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbceqg  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )

Proof of Theorem sbceqg
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2994 . . 3  |-  ( z  =  A  ->  ( [ z  /  x ] B  =  C  <->  [. A  /  x ]. B  =  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, 5eqeq12d 2297 . . 3  |-  ( z  =  A  ->  ( { y  |  [
z  /  x ]
y  e.  B }  =  { y  |  [
z  /  x ]
y  e.  C }  <->  { y  |  [. A  /  x ]. y  e.  B }  =  {
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, 10nfeq 2426 . . . 4  |-  F/ x { y  |  [
z  /  x ]
y  e.  B }  =  { 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, 13eqeq12d 2297 . . . 4  |-  ( x  =  z  ->  ( B  =  C  <->  { y  |  [ z  /  x ] y  e.  B }  =  { y  |  [ z  /  x ] y  e.  C } ) )
1511, 14sbie 1978 . . 3  |-  ( [ z  /  x ] B  =  C  <->  { y  |  [ z  /  x ] y  e.  B }  =  { y  |  [ z  /  x ] y  e.  C } )
161, 6, 15vtoclbg 2844 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  { y  |  [. A  /  x ]. y  e.  B }  =  {
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, 18eqeq12i 2296 . 2  |-  ( [_ A  /  x ]_ B  =  [_ A  /  x ]_ C  <->  { y  |  [. A  /  x ]. y  e.  B }  =  {
y  |  [. A  /  x ]. y  e.  C } )
2016, 19syl6bbr 254 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ 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:  sbcne12g  3099  sbceq1g  3101  sbceq2g  3103  csbfv12gALT  5536  sbcfun  27985  onfrALTlem5  28307  onfrALTlem4  28308  csbeq2g  28315  csbingVD  28660  onfrALTlem5VD  28661  onfrALTlem4VD  28662  csbeq2gVD  28668  csbsngVD  28669  csbunigVD  28674  csbfv12gALTVD  28675  cdlemk42  31130
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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