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Theorem sbcung 25117
Description: Distribution of class substitution over union of two classes. (Contributed by Drahflow, 23-Sep-2015.) (Revised by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
sbcung  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( C  u.  D )  =  ( [_ A  /  x ]_ C  u.  [_ A  /  x ]_ D
) )
Distinct variable group:    x, A
Allowed substitution hints:    C( x)    D( x)

Proof of Theorem sbcung
StepHypRef Expression
1 nfcsb1v 3276 . . . 4  |-  F/_ x [_ A  /  x ]_ C
2 nfcsb1v 3276 . . . 4  |-  F/_ x [_ A  /  x ]_ D
31, 2nfun 3496 . . 3  |-  F/_ x
( [_ A  /  x ]_ C  u.  [_ A  /  x ]_ D )
43a1i 11 . 2  |-  ( A  e.  _V  ->  F/_ x
( [_ A  /  x ]_ C  u.  [_ A  /  x ]_ D ) )
5 csbeq1a 3252 . . 3  |-  ( x  =  A  ->  C  =  [_ A  /  x ]_ C )
6 csbeq1a 3252 . . 3  |-  ( x  =  A  ->  D  =  [_ A  /  x ]_ D )
75, 6uneq12d 3495 . 2  |-  ( x  =  A  ->  ( C  u.  D )  =  ( [_ A  /  x ]_ C  u.  [_ A  /  x ]_ D ) )
84, 7csbiegf 3284 1  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( C  u.  D )  =  ( [_ A  /  x ]_ C  u.  [_ A  /  x ]_ D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   F/_wnfc 2559   _Vcvv 2949   [_csb 3244    u. cun 3311
This theorem is referenced by:  sbcuni  25118
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2951  df-sbc 3155  df-csb 3245  df-un 3318
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