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Theorem reuss 3449
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 21 . . . 4  |-  ( x  e.  A  ->  ( ph  ->  ph ) )
21rgen 2608 . . 3  |-  A. x  e.  A  ( ph  ->  ph )
3 reuss2 3448 . . 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 662 . 2  |-  ( ( A  C_  B  /\  ( E. x  e.  A  ph 
/\  E! x  e.  B  ph ) )  ->  E! x  e.  A  ph )
543impb 1147 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 358    /\ w3a 934    e. wcel 1684   A.wral 2543   E.wrex 2544   E!wreu 2545    C_ wss 3152
This theorem is referenced by:  riotass  6333  adjbdln  22663
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-ral 2548  df-rex 2549  df-reu 2550  df-in 3159  df-ss 3166
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