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Theorem reupick2 3627
Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Assertion
Ref Expression
reupick2  |-  ( ( ( A. x  e.  A  ( ps  ->  ph )  /\  E. x  e.  A  ps  /\  E! x  e.  A  ph )  /\  x  e.  A
)  ->  ( ph  <->  ps ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem reupick2
StepHypRef Expression
1 ancr 533 . . . . . 6  |-  ( ( ps  ->  ph )  -> 
( ps  ->  ( ph  /\  ps ) ) )
21ralimi 2781 . . . . 5  |-  ( A. x  e.  A  ( ps  ->  ph )  ->  A. x  e.  A  ( ps  ->  ( ph  /\  ps ) ) )
3 rexim 2810 . . . . 5  |-  ( A. x  e.  A  ( ps  ->  ( ph  /\  ps ) )  ->  ( E. x  e.  A  ps  ->  E. x  e.  A  ( ph  /\  ps )
) )
42, 3syl 16 . . . 4  |-  ( A. x  e.  A  ( ps  ->  ph )  ->  ( E. x  e.  A  ps  ->  E. x  e.  A  ( ph  /\  ps )
) )
5 reupick3 3626 . . . . . 6  |-  ( ( E! x  e.  A  ph 
/\  E. x  e.  A  ( ph  /\  ps )  /\  x  e.  A
)  ->  ( ph  ->  ps ) )
653exp 1152 . . . . 5  |-  ( E! x  e.  A  ph  ->  ( E. x  e.  A  ( ph  /\  ps )  ->  ( x  e.  A  ->  ( ph  ->  ps ) ) ) )
76com12 29 . . . 4  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  ( E! x  e.  A  ph  ->  (
x  e.  A  -> 
( ph  ->  ps )
) ) )
84, 7syl6 31 . . 3  |-  ( A. x  e.  A  ( ps  ->  ph )  ->  ( E. x  e.  A  ps  ->  ( E! x  e.  A  ph  ->  (
x  e.  A  -> 
( ph  ->  ps )
) ) ) )
983imp1 1166 . 2  |-  ( ( ( A. x  e.  A  ( ps  ->  ph )  /\  E. x  e.  A  ps  /\  E! x  e.  A  ph )  /\  x  e.  A
)  ->  ( ph  ->  ps ) )
10 rsp 2766 . . . 4  |-  ( A. x  e.  A  ( ps  ->  ph )  ->  (
x  e.  A  -> 
( ps  ->  ph )
) )
11103ad2ant1 978 . . 3  |-  ( ( A. x  e.  A  ( ps  ->  ph )  /\  E. x  e.  A  ps  /\  E! x  e.  A  ph )  -> 
( x  e.  A  ->  ( ps  ->  ph )
) )
1211imp 419 . 2  |-  ( ( ( A. x  e.  A  ( ps  ->  ph )  /\  E. x  e.  A  ps  /\  E! x  e.  A  ph )  /\  x  e.  A
)  ->  ( ps  ->  ph ) )
139, 12impbid 184 1  |-  ( ( ( A. x  e.  A  ( ps  ->  ph )  /\  E. x  e.  A  ps  /\  E! x  e.  A  ph )  /\  x  e.  A
)  ->  ( ph  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1725   A.wral 2705   E.wrex 2706   E!wreu 2707
This theorem is referenced by:  grpoidval  21804  grpoidinv2  21806  grpoinv  21815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-ral 2710  df-rex 2711  df-reu 2712
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