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Theorem reuun1 3463
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 3351 . 2  |-  A  C_  ( A  u.  B
)
2 orc 374 . . 3  |-  ( ph  ->  ( ph  \/  ps ) )
32rgenw 2623 . 2  |-  A. x  e.  A  ( ph  ->  ( ph  \/  ps ) )
4 reuss2 3461 . 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 663 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 357    /\ wa 358   A.wral 2556   E.wrex 2557   E!wreu 2558    u. cun 3163    C_ wss 3165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563  df-v 2803  df-un 3170  df-in 3172  df-ss 3179
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