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Theorem rabun2 3276
Description: Abstraction restricted to a union. (Contributed by Stefan O'Rear, 5-Feb-2015.)
Assertion
Ref Expression
rabun2  |-  { x  e.  ( A  u.  B
)  |  ph }  =  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ph } )

Proof of Theorem rabun2
StepHypRef Expression
1 df-rab 2368 . 2  |-  { x  e.  ( A  u.  B
)  |  ph }  =  { x  |  ( x  e.  ( A  u.  B )  /\  ph ) }
2 df-rab 2368 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
3 df-rab 2368 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
42, 3uneq12i 3150 . . 3  |-  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ph }
)  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  ph ) } )
5 elun 3139 . . . . . . 7  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
65anbi1i 446 . . . . . 6  |-  ( ( x  e.  ( A  u.  B )  /\  ph )  <->  ( ( x  e.  A  \/  x  e.  B )  /\  ph ) )
7 andir 768 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  ph )  <->  ( (
x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph ) ) )
86, 7bitri 182 . . . . 5  |-  ( ( x  e.  ( A  u.  B )  /\  ph )  <->  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph )
) )
98abbii 2203 . . . 4  |-  { x  |  ( x  e.  ( A  u.  B
)  /\  ph ) }  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph ) ) }
10 unab 3264 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  ph ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph ) ) }
119, 10eqtr4i 2111 . . 3  |-  { x  |  ( x  e.  ( A  u.  B
)  /\  ph ) }  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  ph ) } )
124, 11eqtr4i 2111 . 2  |-  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ph }
)  =  { x  |  ( x  e.  ( A  u.  B
)  /\  ph ) }
131, 12eqtr4i 2111 1  |-  { x  e.  ( A  u.  B
)  |  ph }  =  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ph } )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    \/ wo 664    = wceq 1289    e. wcel 1438   {cab 2074   {crab 2363    u. cun 2995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-un 3001
This theorem is referenced by:  ssfirab  6622
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