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Theorem reupick2 3781
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 547 . . . . . 6  |-  ( ( ps  ->  ph )  -> 
( ps  ->  ( ph  /\  ps ) ) )
21ralimi 2847 . . . . 5  |-  ( A. x  e.  A  ( ps  ->  ph )  ->  A. x  e.  A  ( ps  ->  ( ph  /\  ps ) ) )
3 rexim 2919 . . . . 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 3780 . . . . . 6  |-  ( ( E! x  e.  A  ph 
/\  E. x  e.  A  ( ph  /\  ps )  /\  x  e.  A
)  ->  ( ph  ->  ps ) )
653exp 1193 . . . . 5  |-  ( E! x  e.  A  ph  ->  ( E. x  e.  A  ( ph  /\  ps )  ->  ( x  e.  A  ->  ( ph  ->  ps ) ) ) )
76com12 31 . . . 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 1207 . 2  |-  ( ( ( A. x  e.  A  ( ps  ->  ph )  /\  E. x  e.  A  ps  /\  E! x  e.  A  ph )  /\  x  e.  A
)  ->  ( ph  ->  ps ) )
10 rsp 2820 . . . 4  |-  ( A. x  e.  A  ( ps  ->  ph )  ->  (
x  e.  A  -> 
( ps  ->  ph )
) )
11103ad2ant1 1015 . . 3  |-  ( ( A. x  e.  A  ( ps  ->  ph )  /\  E. x  e.  A  ps  /\  E! x  e.  A  ph )  -> 
( x  e.  A  ->  ( ps  ->  ph )
) )
1211imp 427 . 2  |-  ( ( ( A. x  e.  A  ( ps  ->  ph )  /\  E. x  e.  A  ps  /\  E! x  e.  A  ph )  /\  x  e.  A
)  ->  ( ps  ->  ph ) )
139, 12impbid 191 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    e. wcel 1823   A.wral 2804   E.wrex 2805   E!wreu 2806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859  ax-13 2004
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973  df-ex 1618  df-nf 1622  df-eu 2288  df-mo 2289  df-ral 2809  df-rex 2810  df-reu 2811
This theorem is referenced by:  grpoidval  25416  grpoidinv2  25418  grpoinv  25427
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