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Theorem reuun2 3568
Description: Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)
Assertion
Ref Expression
reuun2  |-  ( -. 
E. x  e.  B  ph 
->  ( E! x  e.  ( A  u.  B
) ph  <->  E! x  e.  A  ph ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem reuun2
StepHypRef Expression
1 df-rex 2656 . . 3  |-  ( E. x  e.  B  ph  <->  E. x ( x  e.  B  /\  ph )
)
2 euor2 2307 . . 3  |-  ( -. 
E. x ( x  e.  B  /\  ph )  ->  ( E! x
( ( x  e.  B  /\  ph )  \/  ( x  e.  A  /\  ph ) )  <->  E! x
( x  e.  A  /\  ph ) ) )
31, 2sylnbi 298 . 2  |-  ( -. 
E. x  e.  B  ph 
->  ( E! x ( ( x  e.  B  /\  ph )  \/  (
x  e.  A  /\  ph ) )  <->  E! x
( x  e.  A  /\  ph ) ) )
4 df-reu 2657 . . 3  |-  ( E! x  e.  ( A  u.  B ) ph  <->  E! x ( x  e.  ( A  u.  B
)  /\  ph ) )
5 elun 3432 . . . . . 6  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
65anbi1i 677 . . . . 5  |-  ( ( x  e.  ( A  u.  B )  /\  ph )  <->  ( ( x  e.  A  \/  x  e.  B )  /\  ph ) )
7 andir 839 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  ph )  <->  ( (
x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph ) ) )
8 orcom 377 . . . . . 6  |-  ( ( ( x  e.  A  /\  ph )  \/  (
x  e.  B  /\  ph ) )  <->  ( (
x  e.  B  /\  ph )  \/  ( x  e.  A  /\  ph ) ) )
97, 8bitri 241 . . . . 5  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  ph )  <->  ( (
x  e.  B  /\  ph )  \/  ( x  e.  A  /\  ph ) ) )
106, 9bitri 241 . . . 4  |-  ( ( x  e.  ( A  u.  B )  /\  ph )  <->  ( ( x  e.  B  /\  ph )  \/  ( x  e.  A  /\  ph )
) )
1110eubii 2248 . . 3  |-  ( E! x ( x  e.  ( A  u.  B
)  /\  ph )  <->  E! x
( ( x  e.  B  /\  ph )  \/  ( x  e.  A  /\  ph ) ) )
124, 11bitri 241 . 2  |-  ( E! x  e.  ( A  u.  B ) ph  <->  E! x ( ( x  e.  B  /\  ph )  \/  ( x  e.  A  /\  ph )
) )
13 df-reu 2657 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
143, 12, 133bitr4g 280 1  |-  ( -. 
E. x  e.  B  ph 
->  ( E! x  e.  ( A  u.  B
) ph  <->  E! x  e.  A  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359   E.wex 1547    e. wcel 1717   E!weu 2239   E.wrex 2651   E!wreu 2652    u. cun 3262
This theorem is referenced by:  hdmap14lem4a  31990
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-rex 2656  df-reu 2657  df-v 2902  df-un 3269
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