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Theorem reupick 3452
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 3174 . . 3  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21ad2antrr 706 . 2  |-  ( ( ( A  C_  B  /\  ( E. x  e.  A  ph  /\  E! x  e.  B  ph )
)  /\  ph )  -> 
( x  e.  A  ->  x  e.  B ) )
3 df-rex 2549 . . . . . 6  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
4 df-reu 2550 . . . . . 6  |-  ( E! x  e.  B  ph  <->  E! x ( x  e.  B  /\  ph )
)
53, 4anbi12i 678 . . . . 5  |-  ( ( E. x  e.  A  ph 
/\  E! x  e.  B  ph )  <->  ( E. x ( x  e.  A  /\  ph )  /\  E! x ( x  e.  B  /\  ph ) ) )
61ancrd 537 . . . . . . . . . . 11  |-  ( A 
C_  B  ->  (
x  e.  A  -> 
( x  e.  B  /\  x  e.  A
) ) )
76anim1d 547 . . . . . . . . . 10  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  ->  (
( x  e.  B  /\  x  e.  A
)  /\  ph ) ) )
8 an32 773 . . . . . . . . . 10  |-  ( ( ( x  e.  B  /\  x  e.  A
)  /\  ph )  <->  ( (
x  e.  B  /\  ph )  /\  x  e.  A ) )
97, 8syl6ib 217 . . . . . . . . 9  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  ->  (
( x  e.  B  /\  ph )  /\  x  e.  A ) ) )
109eximdv 1608 . . . . . . . 8  |-  ( A 
C_  B  ->  ( E. x ( x  e.  A  /\  ph )  ->  E. x ( ( x  e.  B  /\  ph )  /\  x  e.  A ) ) )
11 eupick 2206 . . . . . . . . 9  |-  ( ( E! x ( x  e.  B  /\  ph )  /\  E. x ( ( x  e.  B  /\  ph )  /\  x  e.  A ) )  -> 
( ( x  e.  B  /\  ph )  ->  x  e.  A ) )
1211ex 423 . . . . . . . 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 66 . . . . . . 7  |-  ( A 
C_  B  ->  ( E! x ( x  e.  B  /\  ph )  ->  ( E. x ( x  e.  A  /\  ph )  ->  ( (
x  e.  B  /\  ph )  ->  x  e.  A ) ) ) )
1413com23 72 . . . . . 6  |-  ( A 
C_  B  ->  ( E. x ( x  e.  A  /\  ph )  ->  ( E! x ( x  e.  B  /\  ph )  ->  ( (
x  e.  B  /\  ph )  ->  x  e.  A ) ) ) )
1514imp32 422 . . . . 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 461 . . . 4  |-  ( ( A  C_  B  /\  ( E. x  e.  A  ph 
/\  E! x  e.  B  ph ) )  ->  ( ( x  e.  B  /\  ph )  ->  x  e.  A
) )
1716exp3acom23 1362 . . 3  |-  ( ( A  C_  B  /\  ( E. x  e.  A  ph 
/\  E! x  e.  B  ph ) )  ->  ( ph  ->  ( x  e.  B  ->  x  e.  A )
) )
1817imp 418 . 2  |-  ( ( ( A  C_  B  /\  ( E. x  e.  A  ph  /\  E! x  e.  B  ph )
)  /\  ph )  -> 
( x  e.  B  ->  x  e.  A ) )
192, 18impbid 183 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    <-> wb 176    /\ wa 358   E.wex 1528    e. wcel 1684   E!weu 2143   E.wrex 2544   E!wreu 2545    C_ wss 3152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-rex 2549  df-reu 2550  df-in 3159  df-ss 3166
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