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Theorem reuss 3755
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 26 . . . 4  |-  ( x  e.  A  ->  ( ph  ->  ph ) )
21rgen 2786 . . 3  |-  A. x  e.  A  ( ph  ->  ph )
3 reuss2 3754 . . 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 686 . 2  |-  ( ( A  C_  B  /\  ( E. x  e.  A  ph 
/\  E! x  e.  B  ph ) )  ->  E! x  e.  A  ph )
543impb 1202 1  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  E! x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 983    e. wcel 1869   A.wral 2776   E.wrex 2777   E!wreu 2778    C_ wss 3437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-ral 2781  df-rex 2782  df-reu 2783  df-in 3444  df-ss 3451
This theorem is referenced by:  euelss  3761  riotass  6292  adjbdln  27728
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