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Theorem reupick 3299
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 3033 . . 3  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21ad2antrr 473 . 2  |-  ( ( ( A  C_  B  /\  ( E. x  e.  A  ph  /\  E! x  e.  B  ph )
)  /\  ph )  -> 
( x  e.  A  ->  x  e.  B ) )
3 df-rex 2376 . . . . . 6  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
4 df-reu 2377 . . . . . 6  |-  ( E! x  e.  B  ph  <->  E! x ( x  e.  B  /\  ph )
)
53, 4anbi12i 449 . . . . 5  |-  ( ( E. x  e.  A  ph 
/\  E! x  e.  B  ph )  <->  ( E. x ( x  e.  A  /\  ph )  /\  E! x ( x  e.  B  /\  ph ) ) )
61ancrd 320 . . . . . . . . . . 11  |-  ( A 
C_  B  ->  (
x  e.  A  -> 
( x  e.  B  /\  x  e.  A
) ) )
76anim1d 330 . . . . . . . . . 10  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  ->  (
( x  e.  B  /\  x  e.  A
)  /\  ph ) ) )
8 an32 530 . . . . . . . . . 10  |-  ( ( ( x  e.  B  /\  x  e.  A
)  /\  ph )  <->  ( (
x  e.  B  /\  ph )  /\  x  e.  A ) )
97, 8syl6ib 160 . . . . . . . . 9  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  ->  (
( x  e.  B  /\  ph )  /\  x  e.  A ) ) )
109eximdv 1815 . . . . . . . 8  |-  ( A 
C_  B  ->  ( E. x ( x  e.  A  /\  ph )  ->  E. x ( ( x  e.  B  /\  ph )  /\  x  e.  A ) ) )
11 eupick 2034 . . . . . . . . 9  |-  ( ( E! x ( x  e.  B  /\  ph )  /\  E. x ( ( x  e.  B  /\  ph )  /\  x  e.  A ) )  -> 
( ( x  e.  B  /\  ph )  ->  x  e.  A ) )
1211ex 114 . . . . . . . 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 72 . . . . . . 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 254 . . . . 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 282 . . . 4  |-  ( ( A  C_  B  /\  ( E. x  e.  A  ph 
/\  E! x  e.  B  ph ) )  ->  ( ( x  e.  B  /\  ph )  ->  x  e.  A
) )
1716expcomd 1382 . . 3  |-  ( ( A  C_  B  /\  ( E. x  e.  A  ph 
/\  E! x  e.  B  ph ) )  ->  ( ph  ->  ( x  e.  B  ->  x  e.  A )
) )
1817imp 123 . 2  |-  ( ( ( A  C_  B  /\  ( E. x  e.  A  ph  /\  E! x  e.  B  ph )
)  /\  ph )  -> 
( x  e.  B  ->  x  e.  A ) )
192, 18impbid 128 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 set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   E.wex 1433    e. wcel 1445   E!weu 1955   E.wrex 2371   E!wreu 2372    C_ wss 3013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-rex 2376  df-reu 2377  df-in 3019  df-ss 3026
This theorem is referenced by:  supelti  6777
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