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Theorem reuun1 3358
Description: Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
Assertion
Ref Expression
reuun1  |-  ( ( E. x  e.  A  ph 
/\  E! x  e.  ( A  u.  B
) ( ph  \/  ps ) )  ->  E! x  e.  A  ph )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem reuun1
StepHypRef Expression
1 ssun1 3239 . 2  |-  A  C_  ( A  u.  B
)
2 orc 701 . . 3  |-  ( ph  ->  ( ph  \/  ps ) )
32rgenw 2487 . 2  |-  A. x  e.  A  ( ph  ->  ( ph  \/  ps ) )
4 reuss2 3356 . 2  |-  ( ( ( A  C_  ( A  u.  B )  /\  A. x  e.  A  ( ph  ->  ( ph  \/  ps ) ) )  /\  ( E. x  e.  A  ph  /\  E! x  e.  ( A  u.  B ) ( ph  \/  ps ) ) )  ->  E! x  e.  A  ph )
51, 3, 4mpanl12 432 1  |-  ( ( E. x  e.  A  ph 
/\  E! x  e.  ( A  u.  B
) ( ph  \/  ps ) )  ->  E! x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 697   A.wral 2416   E.wrex 2417   E!wreu 2418    u. cun 3069    C_ wss 3071
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-v 2688  df-un 3075  df-in 3077  df-ss 3084
This theorem is referenced by: (None)
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