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Theorem reuss 3558
Description: Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)
Assertion
Ref Expression
reuss  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  E! x  e.  A  ph )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem reuss
StepHypRef Expression
1 idd 22 . . . 4  |-  ( x  e.  A  ->  ( ph  ->  ph ) )
21rgen 2707 . . 3  |-  A. x  e.  A  ( ph  ->  ph )
3 reuss2 3557 . . 3  |-  ( ( ( A  C_  B  /\  A. x  e.  A  ( ph  ->  ph ) )  /\  ( E. x  e.  A  ph  /\  E! x  e.  B  ph )
)  ->  E! x  e.  A  ph )
42, 3mpanl2 663 . 2  |-  ( ( A  C_  B  /\  ( E. x  e.  A  ph 
/\  E! x  e.  B  ph ) )  ->  E! x  e.  A  ph )
543impb 1149 1  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  E! x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1717   A.wral 2642   E.wrex 2643   E!wreu 2644    C_ wss 3256
This theorem is referenced by:  riotass  6507  adjbdln  23427
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-ral 2647  df-rex 2648  df-reu 2649  df-in 3263  df-ss 3270
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