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Theorem reuun2 3616
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 2703 . . 3  |-  ( E. x  e.  B  ph  <->  E. x ( x  e.  B  /\  ph )
)
2 euor2 2348 . . 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 2704 . . 3  |-  ( E! x  e.  ( A  u.  B ) ph  <->  E! x ( x  e.  ( A  u.  B
)  /\  ph ) )
5 elun 3480 . . . . . 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 2289 . . 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 2704 . 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 1550    e. wcel 1725   E!weu 2280   E.wrex 2698   E!wreu 2699    u. cun 3310
This theorem is referenced by:  hdmap14lem4a  32573
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-reu 2704  df-v 2950  df-un 3317
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