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Theorem sbcung 24020
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 3113 . . . 4  |-  F/_ x [_ A  /  x ]_ C
2 nfcsb1v 3113 . . . 4  |-  F/_ x [_ A  /  x ]_ D
31, 2nfun 3331 . . 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 3089 . . 3  |-  ( x  =  A  ->  C  =  [_ A  /  x ]_ C )
6 csbeq1a 3089 . . 3  |-  ( x  =  A  ->  D  =  [_ A  /  x ]_ D )
75, 6uneq12d 3330 . 2  |-  ( x  =  A  ->  ( C  u.  D )  =  ( [_ A  /  x ]_ C  u.  [_ A  /  x ]_ D ) )
84, 7csbiegf 3121 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 1623    e. wcel 1684   F/_wnfc 2406   _Vcvv 2788   [_csb 3081    u. cun 3150
This theorem is referenced by:  sbcuni  24021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-csb 3082  df-un 3157
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