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Theorem reuun1 3780
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 3667 . 2  |-  A  C_  ( A  u.  B
)
2 orc 385 . . 3  |-  ( ph  ->  ( ph  \/  ps ) )
32rgenw 2825 . 2  |-  A. x  e.  A  ( ph  ->  ( ph  \/  ps ) )
4 reuss2 3778 . 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 682 1  |-  ( ( E. x  e.  A  ph 
/\  E! x  e.  ( A  u.  B
) ( ph  \/  ps ) )  ->  E! x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369   A.wral 2814   E.wrex 2815   E!wreu 2816    u. cun 3474    C_ wss 3476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-reu 2821  df-v 3115  df-un 3481  df-in 3483  df-ss 3490
This theorem is referenced by: (None)
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