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Theorem reuun2 3733
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 2801 . . 3  |-  ( E. x  e.  B  ph  <->  E. x ( x  e.  B  /\  ph )
)
2 euor2 2319 . . 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 306 . 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 2802 . . 3  |-  ( E! x  e.  ( A  u.  B ) ph  <->  E! x ( x  e.  ( A  u.  B
)  /\  ph ) )
5 elun 3597 . . . . . 6  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
65anbi1i 695 . . . . 5  |-  ( ( x  e.  ( A  u.  B )  /\  ph )  <->  ( ( x  e.  A  \/  x  e.  B )  /\  ph ) )
7 andir 863 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  ph )  <->  ( (
x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph ) ) )
8 orcom 387 . . . . . 6  |-  ( ( ( x  e.  A  /\  ph )  \/  (
x  e.  B  /\  ph ) )  <->  ( (
x  e.  B  /\  ph )  \/  ( x  e.  A  /\  ph ) ) )
97, 8bitri 249 . . . . 5  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  ph )  <->  ( (
x  e.  B  /\  ph )  \/  ( x  e.  A  /\  ph ) ) )
106, 9bitri 249 . . . 4  |-  ( ( x  e.  ( A  u.  B )  /\  ph )  <->  ( ( x  e.  B  /\  ph )  \/  ( x  e.  A  /\  ph )
) )
1110eubii 2285 . . 3  |-  ( E! x ( x  e.  ( A  u.  B
)  /\  ph )  <->  E! x
( ( x  e.  B  /\  ph )  \/  ( x  e.  A  /\  ph ) ) )
124, 11bitri 249 . 2  |-  ( E! x  e.  ( A  u.  B ) ph  <->  E! x ( ( x  e.  B  /\  ph )  \/  ( x  e.  A  /\  ph )
) )
13 df-reu 2802 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
143, 12, 133bitr4g 288 1  |-  ( -. 
E. x  e.  B  ph 
->  ( E! x  e.  ( A  u.  B
) ph  <->  E! x  e.  A  ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369   E.wex 1587    e. wcel 1758   E!weu 2260   E.wrex 2796   E!wreu 2797    u. cun 3426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rex 2801  df-reu 2802  df-v 3072  df-un 3433
This theorem is referenced by:  hdmap14lem4a  35827
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