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Theorem sbcung 24035
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 3126 . . . 4  |-  F/_ x [_ A  /  x ]_ C
2 nfcsb1v 3126 . . . 4  |-  F/_ x [_ A  /  x ]_ D
31, 2nfun 3344 . . 3  |-  F/_ x
( [_ A  /  x ]_ C  u.  [_ A  /  x ]_ D )
43a1i 10 . 2  |-  ( A  e.  _V  ->  F/_ x
( [_ A  /  x ]_ C  u.  [_ A  /  x ]_ D ) )
5 csbeq1a 3102 . . 3  |-  ( x  =  A  ->  C  =  [_ A  /  x ]_ C )
6 csbeq1a 3102 . . 3  |-  ( x  =  A  ->  D  =  [_ A  /  x ]_ D )
75, 6uneq12d 3343 . 2  |-  ( x  =  A  ->  ( C  u.  D )  =  ( [_ A  /  x ]_ C  u.  [_ A  /  x ]_ D ) )
84, 7csbiegf 3134 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 1632    e. wcel 1696   F/_wnfc 2419   _Vcvv 2801   [_csb 3094    u. cun 3163
This theorem is referenced by:  sbcuni  24036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005  df-csb 3095  df-un 3170
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