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Theorem reuun1 3625
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 3512 . 2  |-  A  C_  ( A  u.  B
)
2 orc 376 . . 3  |-  ( ph  ->  ( ph  \/  ps ) )
32rgenw 2775 . 2  |-  A. x  e.  A  ( ph  ->  ( ph  \/  ps ) )
4 reuss2 3623 . 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 665 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    \/ wo 359    /\ wa 360   A.wral 2707   E.wrex 2708   E!wreu 2709    u. cun 3320    C_ wss 3322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-reu 2714  df-v 2960  df-un 3327  df-in 3329  df-ss 3336
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