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Theorem reuun1 3591
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 3478 . 2  |-  A  C_  ( A  u.  B
)
2 orc 375 . . 3  |-  ( ph  ->  ( ph  \/  ps ) )
32rgenw 2741 . 2  |-  A. x  e.  A  ( ph  ->  ( ph  \/  ps ) )
4 reuss2 3589 . 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 664 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 358    /\ wa 359   A.wral 2674   E.wrex 2675   E!wreu 2676    u. cun 3286    C_ wss 3288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-reu 2681  df-v 2926  df-un 3293  df-in 3295  df-ss 3302
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