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Theorem sbceqg 3825
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 3334 . . 3  |-  ( z  =  A  ->  ( [ z  /  x ] B  =  C  <->  [. A  /  x ]. B  =  C )
)
2 dfsbcq2 3334 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] y  e.  B  <->  [. A  /  x ]. y  e.  B )
)
32abbidv 2603 . . . 4  |-  ( z  =  A  ->  { y  |  [ z  /  x ] y  e.  B }  =  { y  |  [. A  /  x ]. y  e.  B } )
4 dfsbcq2 3334 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] y  e.  C  <->  [. A  /  x ]. y  e.  C )
)
54abbidv 2603 . . . 4  |-  ( z  =  A  ->  { y  |  [ z  /  x ] y  e.  C }  =  { y  |  [. A  /  x ]. y  e.  C } )
63, 5eqeq12d 2489 . . 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 2164 . . . . . 6  |-  F/ x [ z  /  x ] y  e.  B
87nfab 2633 . . . . 5  |-  F/_ x { y  |  [
z  /  x ]
y  e.  B }
9 nfs1v 2164 . . . . . 6  |-  F/ x [ z  /  x ] y  e.  C
109nfab 2633 . . . . 5  |-  F/_ x { y  |  [
z  /  x ]
y  e.  C }
118, 10nfeq 2640 . . . 4  |-  F/ x { y  |  [
z  /  x ]
y  e.  B }  =  { y  |  [
z  /  x ]
y  e.  C }
12 sbab 2614 . . . . 5  |-  ( x  =  z  ->  B  =  { y  |  [
z  /  x ]
y  e.  B }
)
13 sbab 2614 . . . . 5  |-  ( x  =  z  ->  C  =  { y  |  [
z  /  x ]
y  e.  C }
)
1412, 13eqeq12d 2489 . . . 4  |-  ( x  =  z  ->  ( B  =  C  <->  { y  |  [ z  /  x ] y  e.  B }  =  { y  |  [ z  /  x ] y  e.  C } ) )
1511, 14sbie 2123 . . 3  |-  ( [ z  /  x ] B  =  C  <->  { y  |  [ z  /  x ] y  e.  B }  =  { y  |  [ z  /  x ] y  e.  C } )
161, 6, 15vtoclbg 3172 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  { y  |  [. A  /  x ]. y  e.  B }  =  {
y  |  [. A  /  x ]. y  e.  C } ) )
17 df-csb 3436 . . 3  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
18 df-csb 3436 . . 3  |-  [_ A  /  x ]_ C  =  { y  |  [. A  /  x ]. y  e.  C }
1917, 18eqeq12i 2487 . 2  |-  ( [_ A  /  x ]_ B  =  [_ A  /  x ]_ C  <->  { y  |  [. A  /  x ]. y  e.  B }  =  {
y  |  [. A  /  x ]. y  e.  C } )
2016, 19syl6bbr 263 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 184    = wceq 1379   [wsb 1711    e. wcel 1767   {cab 2452   [.wsbc 3331   [_csb 3435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-sbc 3332  df-csb 3436
This theorem is referenced by:  sbcne12  3827  sbcne12gOLD  3828  sbceq1g  3830  sbceq2g  3833  csbmpt12  4781  sbcfng  5728  swrdspsleq  12639  sbceqi  30343  onfrALTlem5  32611  onfrALTlem4  32612  csbeq2gOLD  32619  csbfv12gALTOLD  32918  csbingVD  32981  onfrALTlem5VD  32982  onfrALTlem4VD  32983  csbeq2gVD  32989  csbsngVD  32990  csbunigVD  32995  csbfv12gALTVD  32996  cdlemk42  35954
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