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Theorem sbceqg 3065
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 2958 . . 3  |-  ( z  =  A  ->  ( [ z  /  x ] B  =  C  <->  [. A  /  x ]. B  =  C )
)
2 dfsbcq2 2958 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] y  e.  B  <->  [. A  /  x ]. y  e.  B )
)
32abbidv 2288 . . . 4  |-  ( z  =  A  ->  { y  |  [ z  /  x ] y  e.  B }  =  { y  |  [. A  /  x ]. y  e.  B } )
4 dfsbcq2 2958 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] y  e.  C  <->  [. A  /  x ]. y  e.  C )
)
54abbidv 2288 . . . 4  |-  ( z  =  A  ->  { y  |  [ z  /  x ] y  e.  C }  =  { y  |  [. A  /  x ]. y  e.  C } )
63, 5eqeq12d 2185 . . 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 1932 . . . . . 6  |-  F/ x [ z  /  x ] y  e.  B
87nfab 2317 . . . . 5  |-  F/_ x { y  |  [
z  /  x ]
y  e.  B }
9 nfs1v 1932 . . . . . 6  |-  F/ x [ z  /  x ] y  e.  C
109nfab 2317 . . . . 5  |-  F/_ x { y  |  [
z  /  x ]
y  e.  C }
118, 10nfeq 2320 . . . 4  |-  F/ x { y  |  [
z  /  x ]
y  e.  B }  =  { y  |  [
z  /  x ]
y  e.  C }
12 sbab 2298 . . . . 5  |-  ( x  =  z  ->  B  =  { y  |  [
z  /  x ]
y  e.  B }
)
13 sbab 2298 . . . . 5  |-  ( x  =  z  ->  C  =  { y  |  [
z  /  x ]
y  e.  C }
)
1412, 13eqeq12d 2185 . . . 4  |-  ( x  =  z  ->  ( B  =  C  <->  { y  |  [ z  /  x ] y  e.  B }  =  { y  |  [ z  /  x ] y  e.  C } ) )
1511, 14sbie 1784 . . 3  |-  ( [ z  /  x ] B  =  C  <->  { y  |  [ z  /  x ] y  e.  B }  =  { y  |  [ z  /  x ] y  e.  C } )
161, 6, 15vtoclbg 2791 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  { y  |  [. A  /  x ]. y  e.  B }  =  {
y  |  [. A  /  x ]. y  e.  C } ) )
17 df-csb 3050 . . 3  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
18 df-csb 3050 . . 3  |-  [_ A  /  x ]_ C  =  { y  |  [. A  /  x ]. y  e.  C }
1917, 18eqeq12i 2184 . 2  |-  ( [_ A  /  x ]_ B  =  [_ A  /  x ]_ C  <->  { y  |  [. A  /  x ]. y  e.  B }  =  {
y  |  [. A  /  x ]. y  e.  C } )
2016, 19bitr4di 197 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 104    = wceq 1348   [wsb 1755    e. wcel 2141   {cab 2156   [.wsbc 2955   [_csb 3049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050
This theorem is referenced by:  sbcne12g  3067  sbceq1g  3069  sbceq2g  3071  sbcfng  5345  fprodmodd  11604
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