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Theorem csbprg 30252
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 3804 . . 3  |-  [_ C  /  x ]_ ( { A }  u.  { B } )  =  (
[_ C  /  x ]_ { A }  u.  [_ C  /  x ]_ { B } )
2 csbsng 4030 . . . 4  |-  ( C  e.  V  ->  [_ C  /  x ]_ { A }  =  { [_ C  /  x ]_ A }
)
3 csbsng 4030 . . . 4  |-  ( C  e.  V  ->  [_ C  /  x ]_ { B }  =  { [_ C  /  x ]_ B }
)
42, 3uneq12d 3606 . . 3  |-  ( C  e.  V  ->  ( [_ C  /  x ]_ { A }  u.  [_ C  /  x ]_ { B } )  =  ( { [_ C  /  x ]_ A }  u.  { [_ C  /  x ]_ B } ) )
51, 4syl5eq 2503 . 2  |-  ( C  e.  V  ->  [_ C  /  x ]_ ( { A }  u.  { B } )  =  ( { [_ C  /  x ]_ A }  u.  {
[_ C  /  x ]_ B } ) )
6 df-pr 3975 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
76csbeq2i 3783 . 2  |-  [_ C  /  x ]_ { A ,  B }  =  [_ C  /  x ]_ ( { A }  u.  { B } )
8 df-pr 3975 . 2  |-  { [_ C  /  x ]_ A ,  [_ C  /  x ]_ B }  =  ( { [_ C  /  x ]_ A }  u.  {
[_ C  /  x ]_ B } )
95, 7, 83eqtr4g 2516 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 1370    e. wcel 1758   [_csb 3383    u. cun 3421   {csn 3972   {cpr 3974
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-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
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 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-sn 3973  df-pr 3975
This theorem is referenced by:  rusgrasn  30692
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