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Theorem csbopabg 3876
Description: Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
csbopabg  |-  ( A  e.  V  ->  [_ A  /  x ]_ { <. y ,  z >.  |  ph }  =  { <. y ,  z >.  |  [. A  /  x ]. ph }
)
Distinct variable groups:    y, z, A   
x, y, z
Allowed substitution hints:    ph( x, y, z)    A( x)    V( x, y, z)

Proof of Theorem csbopabg
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 csbeq1 2920 . . 3  |-  ( w  =  A  ->  [_ w  /  x ]_ { <. y ,  z >.  |  ph }  =  [_ A  /  x ]_ { <. y ,  z >.  |  ph } )
2 dfsbcq2 2827 . . . 4  |-  ( w  =  A  ->  ( [ w  /  x ] ph  <->  [. A  /  x ]. ph ) )
32opabbidv 3864 . . 3  |-  ( w  =  A  ->  { <. y ,  z >.  |  [
w  /  x ] ph }  =  { <. y ,  z >.  |  [. A  /  x ]. ph }
)
41, 3eqeq12d 2097 . 2  |-  ( w  =  A  ->  ( [_ w  /  x ]_ { <. y ,  z
>.  |  ph }  =  { <. y ,  z
>.  |  [ w  /  x ] ph }  <->  [_ A  /  x ]_ { <. y ,  z
>.  |  ph }  =  { <. y ,  z
>.  |  [. A  /  x ]. ph } ) )
5 vex 2613 . . 3  |-  w  e. 
_V
6 nfs1v 1858 . . . 4  |-  F/ x [ w  /  x ] ph
76nfopab 3866 . . 3  |-  F/_ x { <. y ,  z
>.  |  [ w  /  x ] ph }
8 sbequ12 1696 . . . 4  |-  ( x  =  w  ->  ( ph 
<->  [ w  /  x ] ph ) )
98opabbidv 3864 . . 3  |-  ( x  =  w  ->  { <. y ,  z >.  |  ph }  =  { <. y ,  z >.  |  [
w  /  x ] ph } )
105, 7, 9csbief 2956 . 2  |-  [_ w  /  x ]_ { <. y ,  z >.  |  ph }  =  { <. y ,  z >.  |  [
w  /  x ] ph }
114, 10vtoclg 2667 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ { <. y ,  z >.  |  ph }  =  { <. y ,  z >.  |  [. A  /  x ]. ph }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   [wsb 1687   [.wsbc 2824   [_csb 2917   {copab 3858
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sbc 2825  df-csb 2918  df-opab 3860
This theorem is referenced by:  csbcnvg  4567
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