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Theorem csbopab 4731
Description: Move substitution into a class abstraction. Version of csbopabgALT 4732 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 3401 . . . 4  |-  ( w  =  A  ->  [_ w  /  x ]_ { <. y ,  z >.  |  ph }  =  [_ A  /  x ]_ { <. y ,  z >.  |  ph } )
2 dfsbcq2 3297 . . . . 5  |-  ( w  =  A  ->  ( [ w  /  x ] ph  <->  [. A  /  x ]. ph ) )
32opabbidv 4466 . . . 4  |-  ( w  =  A  ->  { <. y ,  z >.  |  [
w  /  x ] ph }  =  { <. y ,  z >.  |  [. A  /  x ]. ph }
)
41, 3eqeq12d 2476 . . 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 3081 . . . 4  |-  w  e. 
_V
6 nfs1v 2151 . . . . 5  |-  F/ x [ w  /  x ] ph
76nfopab 4468 . . . 4  |-  F/_ x { <. y ,  z
>.  |  [ w  /  x ] ph }
8 sbequ12 1948 . . . . 5  |-  ( x  =  w  ->  ( ph 
<->  [ w  /  x ] ph ) )
98opabbidv 4466 . . . 4  |-  ( x  =  w  ->  { <. y ,  z >.  |  ph }  =  { <. y ,  z >.  |  [
w  /  x ] ph } )
105, 7, 9csbief 3423 . . 3  |-  [_ w  /  x ]_ { <. y ,  z >.  |  ph }  =  { <. y ,  z >.  |  [
w  /  x ] ph }
114, 10vtoclg 3136 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ { <. y ,  z >.  |  ph }  =  { <. y ,  z >.  |  [. A  /  x ]. ph }
)
12 csbprc 3784 . . 3  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ { <. y ,  z
>.  |  ph }  =  (/) )
13 sbcex 3304 . . . . . . 7  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
1413con3i 135 . . . . . 6  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. ph )
1514nexdv 1823 . . . . 5  |-  ( -.  A  e.  _V  ->  -. 
E. z [. A  /  x ]. ph )
1615nexdv 1823 . . . 4  |-  ( -.  A  e.  _V  ->  -. 
E. y E. z [. A  /  x ]. ph )
17 opabn0 4730 . . . . 5  |-  ( {
<. y ,  z >.  |  [. A  /  x ]. ph }  =/=  (/)  <->  E. y E. z [. A  /  x ]. ph )
1817necon1bbii 2716 . . . 4  |-  ( -. 
E. y E. z [. A  /  x ]. ph  <->  { <. y ,  z
>.  |  [. A  /  x ]. ph }  =  (/) )
1916, 18sylib 196 . . 3  |-  ( -.  A  e.  _V  ->  {
<. y ,  z >.  |  [. A  /  x ]. ph }  =  (/) )
2012, 19eqtr4d 2498 . 2  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ { <. y ,  z
>.  |  ph }  =  { <. y ,  z
>.  |  [. A  /  x ]. ph } )
2111, 20pm2.61i 164 1  |-  [_ A  /  x ]_ { <. y ,  z >.  |  ph }  =  { <. y ,  z >.  |  [. A  /  x ]. ph }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1370   E.wex 1587   [wsb 1702    e. wcel 1758   _Vcvv 3078   [.wsbc 3294   [_csb 3398   (/)c0 3748   {copab 4460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-opab 4462
This theorem is referenced by:  csbmpt12  4733  csbcnv  5134
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