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Theorem csbprg 4075
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 3848 . . 3  |-  [_ C  /  x ]_ ( { A }  u.  { B } )  =  (
[_ C  /  x ]_ { A }  u.  [_ C  /  x ]_ { B } )
2 csbsng 4074 . . . 4  |-  ( C  e.  V  ->  [_ C  /  x ]_ { A }  =  { [_ C  /  x ]_ A }
)
3 csbsng 4074 . . . 4  |-  ( C  e.  V  ->  [_ C  /  x ]_ { B }  =  { [_ C  /  x ]_ B }
)
42, 3uneq12d 3645 . . 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 4019 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
76csbeq2i 3832 . 2  |-  [_ C  /  x ]_ { A ,  B }  =  [_ C  /  x ]_ ( { A }  u.  { B } )
8 df-pr 4019 . 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 1398    e. wcel 1823   [_csb 3420    u. cun 3459   {csn 4016   {cpr 4018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-sn 4017  df-pr 4019
This theorem is referenced by:  rusgrasn  25150
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