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Theorem reupick2 3251
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 308 . . . . . 6  |-  ( ( ps  ->  ph )  -> 
( ps  ->  ( ph  /\  ps ) ) )
21ralimi 2401 . . . . 5  |-  ( A. x  e.  A  ( ps  ->  ph )  ->  A. x  e.  A  ( ps  ->  ( ph  /\  ps ) ) )
3 rexim 2430 . . . . 5  |-  ( A. x  e.  A  ( ps  ->  ( ph  /\  ps ) )  ->  ( E. x  e.  A  ps  ->  E. x  e.  A  ( ph  /\  ps )
) )
42, 3syl 14 . . . 4  |-  ( A. x  e.  A  ( ps  ->  ph )  ->  ( E. x  e.  A  ps  ->  E. x  e.  A  ( ph  /\  ps )
) )
5 reupick3 3250 . . . . . 6  |-  ( ( E! x  e.  A  ph 
/\  E. x  e.  A  ( ph  /\  ps )  /\  x  e.  A
)  ->  ( ph  ->  ps ) )
653exp 1114 . . . . 5  |-  ( E! x  e.  A  ph  ->  ( E. x  e.  A  ( ph  /\  ps )  ->  ( x  e.  A  ->  ( ph  ->  ps ) ) ) )
76com12 30 . . . 4  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  ( E! x  e.  A  ph  ->  (
x  e.  A  -> 
( ph  ->  ps )
) ) )
84, 7syl6 33 . . 3  |-  ( A. x  e.  A  ( ps  ->  ph )  ->  ( E. x  e.  A  ps  ->  ( E! x  e.  A  ph  ->  (
x  e.  A  -> 
( ph  ->  ps )
) ) ) )
983imp1 1128 . 2  |-  ( ( ( A. x  e.  A  ( ps  ->  ph )  /\  E. x  e.  A  ps  /\  E! x  e.  A  ph )  /\  x  e.  A
)  ->  ( ph  ->  ps ) )
10 rsp 2386 . . . 4  |-  ( A. x  e.  A  ( ps  ->  ph )  ->  (
x  e.  A  -> 
( ps  ->  ph )
) )
11103ad2ant1 936 . . 3  |-  ( ( A. x  e.  A  ( ps  ->  ph )  /\  E. x  e.  A  ps  /\  E! x  e.  A  ph )  -> 
( x  e.  A  ->  ( ps  ->  ph )
) )
1211imp 119 . 2  |-  ( ( ( A. x  e.  A  ( ps  ->  ph )  /\  E. x  e.  A  ps  /\  E! x  e.  A  ph )  /\  x  e.  A
)  ->  ( ps  ->  ph ) )
139, 12impbid 124 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    /\ wa 101    <-> wb 102    /\ w3a 896    e. wcel 1409   A.wral 2323   E.wrex 2324   E!wreu 2325
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-3an 898  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-ral 2328  df-rex 2329  df-reu 2330
This theorem is referenced by: (None)
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