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Theorem sbceqg 3801
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 3302 . . 3  |-  ( z  =  A  ->  ( [ z  /  x ] B  =  C  <->  [. A  /  x ]. B  =  C )
)
2 dfsbcq2 3302 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] y  e.  B  <->  [. A  /  x ]. y  e.  B )
)
32abbidv 2558 . . . 4  |-  ( z  =  A  ->  { y  |  [ z  /  x ] y  e.  B }  =  { y  |  [. A  /  x ]. y  e.  B } )
4 dfsbcq2 3302 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] y  e.  C  <->  [. A  /  x ]. y  e.  C )
)
54abbidv 2558 . . . 4  |-  ( z  =  A  ->  { y  |  [ z  /  x ] y  e.  C }  =  { y  |  [. A  /  x ]. y  e.  C } )
63, 5eqeq12d 2444 . . 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 2232 . . . . . 6  |-  F/ x [ z  /  x ] y  e.  B
87nfab 2588 . . . . 5  |-  F/_ x { y  |  [
z  /  x ]
y  e.  B }
9 nfs1v 2232 . . . . . 6  |-  F/ x [ z  /  x ] y  e.  C
109nfab 2588 . . . . 5  |-  F/_ x { y  |  [
z  /  x ]
y  e.  C }
118, 10nfeq 2595 . . . 4  |-  F/ x { y  |  [
z  /  x ]
y  e.  B }  =  { y  |  [
z  /  x ]
y  e.  C }
12 sbab 2569 . . . . 5  |-  ( x  =  z  ->  B  =  { y  |  [
z  /  x ]
y  e.  B }
)
13 sbab 2569 . . . . 5  |-  ( x  =  z  ->  C  =  { y  |  [
z  /  x ]
y  e.  C }
)
1412, 13eqeq12d 2444 . . . 4  |-  ( x  =  z  ->  ( B  =  C  <->  { y  |  [ z  /  x ] y  e.  B }  =  { y  |  [ z  /  x ] y  e.  C } ) )
1511, 14sbie 2202 . . 3  |-  ( [ z  /  x ] B  =  C  <->  { y  |  [ z  /  x ] y  e.  B }  =  { y  |  [ z  /  x ] y  e.  C } )
161, 6, 15vtoclbg 3140 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  { y  |  [. A  /  x ]. y  e.  B }  =  {
y  |  [. A  /  x ]. y  e.  C } ) )
17 df-csb 3396 . . 3  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
18 df-csb 3396 . . 3  |-  [_ A  /  x ]_ C  =  { y  |  [. A  /  x ]. y  e.  C }
1917, 18eqeq12i 2442 . 2  |-  ( [_ A  /  x ]_ B  =  [_ A  /  x ]_ C  <->  { y  |  [. A  /  x ]. y  e.  B }  =  {
y  |  [. A  /  x ]. y  e.  C } )
2016, 19syl6bbr 266 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437   [wsb 1786    e. wcel 1868   {cab 2407   [.wsbc 3299   [_csb 3395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-sbc 3300  df-csb 3396
This theorem is referenced by:  sbcne12  3803  sbceq1g  3805  sbceq2g  3807  sbcfng  5739  swrdspsleq  12793  relowlpssretop  31705  poimirlem25  31876  sbceqi  32259  cdlemk42  34424  onfrALTlem5  36763  onfrALTlem4  36764  csbeq2gOLD  36771  csbfv12gALTOLD  37071  csbingVD  37139  onfrALTlem5VD  37140  onfrALTlem4VD  37141  csbeq2gVD  37147  csbsngVD  37148  csbunigVD  37153  csbfv12gALTVD  37154
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