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Theorem rabun2 3406
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 2457 . 2  |-  { x  e.  ( A  u.  B
)  |  ph }  =  { x  |  ( x  e.  ( A  u.  B )  /\  ph ) }
2 df-rab 2457 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
3 df-rab 2457 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
42, 3uneq12i 3279 . . 3  |-  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ph }
)  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  ph ) } )
5 elun 3268 . . . . . . 7  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
65anbi1i 455 . . . . . 6  |-  ( ( x  e.  ( A  u.  B )  /\  ph )  <->  ( ( x  e.  A  \/  x  e.  B )  /\  ph ) )
7 andir 814 . . . . . 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 2286 . . . 4  |-  { x  |  ( x  e.  ( A  u.  B
)  /\  ph ) }  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph ) ) }
10 unab 3394 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  ph ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph ) ) }
119, 10eqtr4i 2194 . . 3  |-  { x  |  ( x  e.  ( A  u.  B
)  /\  ph ) }  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  ph ) } )
124, 11eqtr4i 2194 . 2  |-  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ph }
)  =  { x  |  ( x  e.  ( A  u.  B
)  /\  ph ) }
131, 12eqtr4i 2194 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 703    = wceq 1348    e. wcel 2141   {cab 2156   {crab 2452    u. cun 3119
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-un 3125
This theorem is referenced by:  ssfirab  6911
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