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Theorem reupick 3787
Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)
Assertion
Ref Expression
reupick  |-  ( ( ( A  C_  B  /\  ( E. x  e.  A  ph  /\  E! x  e.  B  ph )
)  /\  ph )  -> 
( x  e.  A  <->  x  e.  B ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem reupick
StepHypRef Expression
1 ssel 3503 . . 3  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21ad2antrr 725 . 2  |-  ( ( ( A  C_  B  /\  ( E. x  e.  A  ph  /\  E! x  e.  B  ph )
)  /\  ph )  -> 
( x  e.  A  ->  x  e.  B ) )
3 df-rex 2823 . . . . . 6  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
4 df-reu 2824 . . . . . 6  |-  ( E! x  e.  B  ph  <->  E! x ( x  e.  B  /\  ph )
)
53, 4anbi12i 697 . . . . 5  |-  ( ( E. x  e.  A  ph 
/\  E! x  e.  B  ph )  <->  ( E. x ( x  e.  A  /\  ph )  /\  E! x ( x  e.  B  /\  ph ) ) )
61ancrd 554 . . . . . . . . . . 11  |-  ( A 
C_  B  ->  (
x  e.  A  -> 
( x  e.  B  /\  x  e.  A
) ) )
76anim1d 564 . . . . . . . . . 10  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  ->  (
( x  e.  B  /\  x  e.  A
)  /\  ph ) ) )
8 an32 796 . . . . . . . . . 10  |-  ( ( ( x  e.  B  /\  x  e.  A
)  /\  ph )  <->  ( (
x  e.  B  /\  ph )  /\  x  e.  A ) )
97, 8syl6ib 226 . . . . . . . . 9  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  ->  (
( x  e.  B  /\  ph )  /\  x  e.  A ) ) )
109eximdv 1686 . . . . . . . 8  |-  ( A 
C_  B  ->  ( E. x ( x  e.  A  /\  ph )  ->  E. x ( ( x  e.  B  /\  ph )  /\  x  e.  A ) ) )
11 eupick 2364 . . . . . . . . 9  |-  ( ( E! x ( x  e.  B  /\  ph )  /\  E. x ( ( x  e.  B  /\  ph )  /\  x  e.  A ) )  -> 
( ( x  e.  B  /\  ph )  ->  x  e.  A ) )
1211ex 434 . . . . . . . 8  |-  ( E! x ( x  e.  B  /\  ph )  ->  ( E. x ( ( x  e.  B  /\  ph )  /\  x  e.  A )  ->  (
( x  e.  B  /\  ph )  ->  x  e.  A ) ) )
1310, 12syl9 71 . . . . . . 7  |-  ( A 
C_  B  ->  ( E! x ( x  e.  B  /\  ph )  ->  ( E. x ( x  e.  A  /\  ph )  ->  ( (
x  e.  B  /\  ph )  ->  x  e.  A ) ) ) )
1413com23 78 . . . . . 6  |-  ( A 
C_  B  ->  ( E. x ( x  e.  A  /\  ph )  ->  ( E! x ( x  e.  B  /\  ph )  ->  ( (
x  e.  B  /\  ph )  ->  x  e.  A ) ) ) )
1514imp32 433 . . . . 5  |-  ( ( A  C_  B  /\  ( E. x ( x  e.  A  /\  ph )  /\  E! x ( x  e.  B  /\  ph ) ) )  -> 
( ( x  e.  B  /\  ph )  ->  x  e.  A ) )
165, 15sylan2b 475 . . . 4  |-  ( ( A  C_  B  /\  ( E. x  e.  A  ph 
/\  E! x  e.  B  ph ) )  ->  ( ( x  e.  B  /\  ph )  ->  x  e.  A
) )
1716expcomd 438 . . 3  |-  ( ( A  C_  B  /\  ( E. x  e.  A  ph 
/\  E! x  e.  B  ph ) )  ->  ( ph  ->  ( x  e.  B  ->  x  e.  A )
) )
1817imp 429 . 2  |-  ( ( ( A  C_  B  /\  ( E. x  e.  A  ph  /\  E! x  e.  B  ph )
)  /\  ph )  -> 
( x  e.  B  ->  x  e.  A ) )
192, 18impbid 191 1  |-  ( ( ( A  C_  B  /\  ( E. x  e.  A  ph  /\  E! x  e.  B  ph )
)  /\  ph )  -> 
( x  e.  A  <->  x  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   E.wex 1596    e. wcel 1767   E!weu 2275   E.wrex 2818   E!wreu 2819    C_ wss 3481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-rex 2823  df-reu 2824  df-in 3488  df-ss 3495
This theorem is referenced by: (None)
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