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Theorem csbopab 4733
Description: Move substitution into a class abstraction. Version of csbopabgALT 4734 without a sethood antecedent but depending on more axioms. (Contributed by NM, 6-Aug-2007.) (Revised by NM, 23-Aug-2018.)
Assertion
Ref Expression
csbopab  |-  [_ 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)

Proof of Theorem csbopab
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3352 . . . 4  |-  ( w  =  A  ->  [_ w  /  x ]_ { <. y ,  z >.  |  ph }  =  [_ A  /  x ]_ { <. y ,  z >.  |  ph } )
2 dfsbcq2 3258 . . . . 5  |-  ( w  =  A  ->  ( [ w  /  x ] ph  <->  [. A  /  x ]. ph ) )
32opabbidv 4459 . . . 4  |-  ( w  =  A  ->  { <. y ,  z >.  |  [
w  /  x ] ph }  =  { <. y ,  z >.  |  [. A  /  x ]. ph }
)
41, 3eqeq12d 2486 . . 3  |-  ( w  =  A  ->  ( [_ w  /  x ]_ { <. y ,  z
>.  |  ph }  =  { <. y ,  z
>.  |  [ w  /  x ] ph }  <->  [_ A  /  x ]_ { <. y ,  z
>.  |  ph }  =  { <. y ,  z
>.  |  [. A  /  x ]. ph } ) )
5 vex 3034 . . . 4  |-  w  e. 
_V
6 nfs1v 2286 . . . . 5  |-  F/ x [ w  /  x ] ph
76nfopab 4461 . . . 4  |-  F/_ x { <. y ,  z
>.  |  [ w  /  x ] ph }
8 sbequ12 2098 . . . . 5  |-  ( x  =  w  ->  ( ph 
<->  [ w  /  x ] ph ) )
98opabbidv 4459 . . . 4  |-  ( x  =  w  ->  { <. y ,  z >.  |  ph }  =  { <. y ,  z >.  |  [
w  /  x ] ph } )
105, 7, 9csbief 3374 . . 3  |-  [_ w  /  x ]_ { <. y ,  z >.  |  ph }  =  { <. y ,  z >.  |  [
w  /  x ] ph }
114, 10vtoclg 3093 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ { <. y ,  z >.  |  ph }  =  { <. y ,  z >.  |  [. A  /  x ]. ph }
)
12 csbprc 3774 . . 3  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ { <. y ,  z
>.  |  ph }  =  (/) )
13 sbcex 3265 . . . . . . 7  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
1413con3i 142 . . . . . 6  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. ph )
1514nexdv 1790 . . . . 5  |-  ( -.  A  e.  _V  ->  -. 
E. z [. A  /  x ]. ph )
1615nexdv 1790 . . . 4  |-  ( -.  A  e.  _V  ->  -. 
E. y E. z [. A  /  x ]. ph )
17 opabn0 4732 . . . . 5  |-  ( {
<. y ,  z >.  |  [. A  /  x ]. ph }  =/=  (/)  <->  E. y E. z [. A  /  x ]. ph )
1817necon1bbii 2692 . . . 4  |-  ( -. 
E. y E. z [. A  /  x ]. ph  <->  { <. y ,  z
>.  |  [. A  /  x ]. ph }  =  (/) )
1916, 18sylib 201 . . 3  |-  ( -.  A  e.  _V  ->  {
<. y ,  z >.  |  [. A  /  x ]. ph }  =  (/) )
2012, 19eqtr4d 2508 . 2  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ { <. y ,  z
>.  |  ph }  =  { <. y ,  z
>.  |  [. A  /  x ]. ph } )
2111, 20pm2.61i 169 1  |-  [_ A  /  x ]_ { <. y ,  z >.  |  ph }  =  { <. y ,  z >.  |  [. A  /  x ]. ph }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1452   E.wex 1671   [wsb 1805    e. wcel 1904   _Vcvv 3031   [.wsbc 3255   [_csb 3349   (/)c0 3722   {copab 4453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455
This theorem is referenced by:  csbmpt12  4735  csbcnv  5023
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