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Theorem csbprg 4042
Description: Distribute proper substitution through a pair of classes. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
Assertion
Ref Expression
csbprg  |-  ( C  e.  V  ->  [_ C  /  x ]_ { A ,  B }  =  { [_ C  /  x ]_ A ,  [_ C  /  x ]_ B }
)

Proof of Theorem csbprg
StepHypRef Expression
1 csbun 3809 . . 3  |-  [_ C  /  x ]_ ( { A }  u.  { B } )  =  (
[_ C  /  x ]_ { A }  u.  [_ C  /  x ]_ { B } )
2 csbsng 4041 . . . 4  |-  ( C  e.  V  ->  [_ C  /  x ]_ { A }  =  { [_ C  /  x ]_ A }
)
3 csbsng 4041 . . . 4  |-  ( C  e.  V  ->  [_ C  /  x ]_ { B }  =  { [_ C  /  x ]_ B }
)
42, 3uneq12d 3600 . . 3  |-  ( C  e.  V  ->  ( [_ C  /  x ]_ { A }  u.  [_ C  /  x ]_ { B } )  =  ( { [_ C  /  x ]_ A }  u.  { [_ C  /  x ]_ B } ) )
51, 4syl5eq 2507 . 2  |-  ( C  e.  V  ->  [_ C  /  x ]_ ( { A }  u.  { B } )  =  ( { [_ C  /  x ]_ A }  u.  {
[_ C  /  x ]_ B } ) )
6 df-pr 3982 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
76csbeq2i 3793 . 2  |-  [_ C  /  x ]_ { A ,  B }  =  [_ C  /  x ]_ ( { A }  u.  { B } )
8 df-pr 3982 . 2  |-  { [_ C  /  x ]_ A ,  [_ C  /  x ]_ B }  =  ( { [_ C  /  x ]_ A }  u.  {
[_ C  /  x ]_ B } )
95, 7, 83eqtr4g 2520 1  |-  ( C  e.  V  ->  [_ C  /  x ]_ { A ,  B }  =  { [_ C  /  x ]_ A ,  [_ C  /  x ]_ B }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1454    e. wcel 1897   [_csb 3374    u. cun 3413   {csn 3979   {cpr 3981
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-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-sn 3980  df-pr 3982
This theorem is referenced by:  rusgrasn  25721  csbopg2  31769
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