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Theorem csbopabgALT 4747
Description: Move substitution into a class abstraction. Version of csbopab 4746 with a sethood antecedent but depending on fewer axioms. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbopabgALT  |-  ( 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 csbopabgALT
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3377 . . 3  |-  ( w  =  A  ->  [_ w  /  x ]_ { <. y ,  z >.  |  ph }  =  [_ A  /  x ]_ { <. y ,  z >.  |  ph } )
2 dfsbcq2 3281 . . . 4  |-  ( w  =  A  ->  ( [ w  /  x ] ph  <->  [. A  /  x ]. ph ) )
32opabbidv 4479 . . 3  |-  ( w  =  A  ->  { <. y ,  z >.  |  [
w  /  x ] ph }  =  { <. y ,  z >.  |  [. A  /  x ]. ph }
)
41, 3eqeq12d 2476 . 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 3059 . . 3  |-  w  e. 
_V
6 nfs1v 2276 . . . 4  |-  F/ x [ w  /  x ] ph
76nfopab 4481 . . 3  |-  F/_ x { <. y ,  z
>.  |  [ w  /  x ] ph }
8 sbequ12 2093 . . . 4  |-  ( x  =  w  ->  ( ph 
<->  [ w  /  x ] ph ) )
98opabbidv 4479 . . 3  |-  ( x  =  w  ->  { <. y ,  z >.  |  ph }  =  { <. y ,  z >.  |  [
w  /  x ] ph } )
105, 7, 9csbief 3399 . 2  |-  [_ w  /  x ]_ { <. y ,  z >.  |  ph }  =  { <. y ,  z >.  |  [
w  /  x ] ph }
114, 10vtoclg 3118 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ { <. y ,  z >.  |  ph }  =  { <. y ,  z >.  |  [. A  /  x ]. ph }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1454   [wsb 1807    e. wcel 1897   [.wsbc 3278   [_csb 3374   {copab 4473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-v 3058  df-sbc 3279  df-csb 3375  df-opab 4475
This theorem is referenced by:  csbcnvgALT  5037
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