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Theorem csbopg2 31769
Description: Distribution of class substitution over ordered pairs. This version has no distinct variable conditions and uses a class variable  V instead of the class of all sets  _V. (Contributed by ML, 25-Oct-2020.)
Assertion
Ref Expression
csbopg2  |-  ( A  e.  V  ->  [_ A  /  x ]_ <. C ,  D >.  =  <. [_ A  /  x ]_ C ,  [_ A  /  x ]_ D >. )

Proof of Theorem csbopg2
StepHypRef Expression
1 csbif 3942 . . 3  |-  [_ A  /  x ]_ if ( ( C  e.  _V  /\  D  e.  _V ) ,  { { C } ,  { C ,  D } } ,  (/) )  =  if ( [. A  /  x ]. ( C  e.  _V  /\  D  e.  _V ) ,  [_ A  /  x ]_ { { C } ,  { C ,  D } } ,  [_ A  /  x ]_ (/) )
2 sbcan 3321 . . . . 5  |-  ( [. A  /  x ]. ( C  e.  _V  /\  D  e.  _V )  <->  ( [. A  /  x ]. C  e.  _V  /\  [. A  /  x ]. D  e. 
_V ) )
3 sbcel1g 3787 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. C  e.  _V  <->  [_ A  /  x ]_ C  e.  _V )
)
4 sbcel1g 3787 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. D  e.  _V  <->  [_ A  /  x ]_ D  e.  _V )
)
53, 4anbi12d 722 . . . . 5  |-  ( A  e.  V  ->  (
( [. A  /  x ]. C  e.  _V  /\ 
[. A  /  x ]. D  e.  _V ) 
<->  ( [_ A  /  x ]_ C  e.  _V  /\ 
[_ A  /  x ]_ D  e.  _V ) ) )
62, 5syl5bb 265 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( C  e.  _V  /\  D  e.  _V )  <->  (
[_ A  /  x ]_ C  e.  _V  /\ 
[_ A  /  x ]_ D  e.  _V ) ) )
7 csbprg 4042 . . . . 5  |-  ( A  e.  V  ->  [_ A  /  x ]_ { { C } ,  { C ,  D } }  =  { [_ A  /  x ]_ { C } ,  [_ A  /  x ]_ { C ,  D } } )
8 csbsng 4041 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ { C }  =  { [_ A  /  x ]_ C }
)
9 csbprg 4042 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ { C ,  D }  =  { [_ A  /  x ]_ C ,  [_ A  /  x ]_ D }
)
108, 9preq12d 4071 . . . . 5  |-  ( A  e.  V  ->  { [_ A  /  x ]_ { C } ,  [_ A  /  x ]_ { C ,  D } }  =  { { [_ A  /  x ]_ C } ,  { [_ A  /  x ]_ C ,  [_ A  /  x ]_ D } } )
117, 10eqtrd 2495 . . . 4  |-  ( A  e.  V  ->  [_ A  /  x ]_ { { C } ,  { C ,  D } }  =  { { [_ A  /  x ]_ C } ,  { [_ A  /  x ]_ C ,  [_ A  /  x ]_ D } } )
12 csbconstg 3387 . . . 4  |-  ( A  e.  V  ->  [_ A  /  x ]_ (/)  =  (/) )
136, 11, 12ifbieq12d 3919 . . 3  |-  ( A  e.  V  ->  if ( [. A  /  x ]. ( C  e.  _V  /\  D  e.  _V ) ,  [_ A  /  x ]_ { { C } ,  { C ,  D } } ,  [_ A  /  x ]_ (/) )  =  if ( ( [_ A  /  x ]_ C  e.  _V  /\  [_ A  /  x ]_ D  e. 
_V ) ,  { { [_ A  /  x ]_ C } ,  { [_ A  /  x ]_ C ,  [_ A  /  x ]_ D } } ,  (/) ) )
141, 13syl5eq 2507 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ if ( ( C  e.  _V  /\  D  e.  _V ) ,  { { C } ,  { C ,  D } } ,  (/) )  =  if ( ( [_ A  /  x ]_ C  e.  _V  /\  [_ A  /  x ]_ D  e. 
_V ) ,  { { [_ A  /  x ]_ C } ,  { [_ A  /  x ]_ C ,  [_ A  /  x ]_ D } } ,  (/) ) )
15 dfopif 4176 . . 3  |-  <. C ,  D >.  =  if ( ( C  e.  _V  /\  D  e.  _V ) ,  { { C } ,  { C ,  D } } ,  (/) )
1615csbeq2i 3793 . 2  |-  [_ A  /  x ]_ <. C ,  D >.  =  [_ A  /  x ]_ if ( ( C  e.  _V  /\  D  e.  _V ) ,  { { C } ,  { C ,  D } } ,  (/) )
17 dfopif 4176 . 2  |-  <. [_ A  /  x ]_ C ,  [_ A  /  x ]_ D >.  =  if ( ( [_ A  /  x ]_ C  e. 
_V  /\  [_ A  /  x ]_ D  e.  _V ) ,  { { [_ A  /  x ]_ C } ,  { [_ A  /  x ]_ C ,  [_ A  /  x ]_ D } } ,  (/) )
1814, 16, 173eqtr4g 2520 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ <. C ,  D >.  =  <. [_ A  /  x ]_ C ,  [_ A  /  x ]_ D >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1454    e. wcel 1897   _Vcvv 3056   [.wsbc 3278   [_csb 3374   (/)c0 3742   ifcif 3892   {csn 3979   {cpr 3981   <.cop 3985
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-or 376  df-an 377  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986
This theorem is referenced by:  csbfinxpg  31824
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