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Theorem sbcuni 25130
Description: Distribution of class substitution over union of two classes, inference version. (Contributed by Drahflow, 23-Sep-2015.)
Hypothesis
Ref Expression
sbcun.1  |-  A  e. 
_V
Assertion
Ref Expression
sbcuni  |-  [_ 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 sbcuni
StepHypRef Expression
1 sbcun.1 . 2  |-  A  e. 
_V
2 sbcung 25129 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( C  u.  D )  =  ( [_ A  /  x ]_ C  u.  [_ A  /  x ]_ D
) )
31, 2ax-mp 5 1  |-  [_ A  /  x ]_ ( C  u.  D )  =  ( [_ A  /  x ]_ C  u.  [_ A  /  x ]_ D
)
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   _Vcvv 2958   [_csb 3253    u. cun 3320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-sbc 3164  df-csb 3254  df-un 3327
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