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Theorem csbopabg 4055
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 3044 . . 3  |-  ( w  =  A  ->  [_ w  /  x ]_ { <. y ,  z >.  |  ph }  =  [_ A  /  x ]_ { <. y ,  z >.  |  ph } )
2 dfsbcq2 2950 . . . 4  |-  ( w  =  A  ->  ( [ w  /  x ] ph  <->  [. A  /  x ]. ph ) )
32opabbidv 4043 . . 3  |-  ( w  =  A  ->  { <. y ,  z >.  |  [
w  /  x ] ph }  =  { <. y ,  z >.  |  [. A  /  x ]. ph }
)
41, 3eqeq12d 2179 . 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 2725 . . 3  |-  w  e. 
_V
6 nfs1v 1926 . . . 4  |-  F/ x [ w  /  x ] ph
76nfopab 4045 . . 3  |-  F/_ x { <. y ,  z
>.  |  [ w  /  x ] ph }
8 sbequ12 1758 . . . 4  |-  ( x  =  w  ->  ( ph 
<->  [ w  /  x ] ph ) )
98opabbidv 4043 . . 3  |-  ( x  =  w  ->  { <. y ,  z >.  |  ph }  =  { <. y ,  z >.  |  [
w  /  x ] ph } )
105, 7, 9csbief 3085 . 2  |-  [_ w  /  x ]_ { <. y ,  z >.  |  ph }  =  { <. y ,  z >.  |  [
w  /  x ] ph }
114, 10vtoclg 2782 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 1342   [wsb 1749    e. wcel 2135   [.wsbc 2947   [_csb 3041   {copab 4037
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-sbc 2948  df-csb 3042  df-opab 4039
This theorem is referenced by:  csbcnvg  4783
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