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Theorem rabun2 3386
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 2444 . 2  |-  { x  e.  ( A  u.  B
)  |  ph }  =  { x  |  ( x  e.  ( A  u.  B )  /\  ph ) }
2 df-rab 2444 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
3 df-rab 2444 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
42, 3uneq12i 3259 . . 3  |-  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ph }
)  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  ph ) } )
5 elun 3248 . . . . . . 7  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
65anbi1i 454 . . . . . 6  |-  ( ( x  e.  ( A  u.  B )  /\  ph )  <->  ( ( x  e.  A  \/  x  e.  B )  /\  ph ) )
7 andir 809 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  ph )  <->  ( (
x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph ) ) )
86, 7bitri 183 . . . . 5  |-  ( ( x  e.  ( A  u.  B )  /\  ph )  <->  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph )
) )
98abbii 2273 . . . 4  |-  { x  |  ( x  e.  ( A  u.  B
)  /\  ph ) }  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph ) ) }
10 unab 3374 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  ph ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph ) ) }
119, 10eqtr4i 2181 . . 3  |-  { x  |  ( x  e.  ( A  u.  B
)  /\  ph ) }  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  ph ) } )
124, 11eqtr4i 2181 . 2  |-  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ph }
)  =  { x  |  ( x  e.  ( A  u.  B
)  /\  ph ) }
131, 12eqtr4i 2181 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 103    \/ wo 698    = wceq 1335    e. wcel 2128   {cab 2143   {crab 2439    u. cun 3100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-v 2714  df-un 3106
This theorem is referenced by:  ssfirab  6878
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