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Theorem reupick3 3453
Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
Assertion
Ref Expression
reupick3  |-  ( ( E! x  e.  A  ph 
/\  E. x  e.  A  ( ph  /\  ps )  /\  x  e.  A
)  ->  ( ph  ->  ps ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem reupick3
StepHypRef Expression
1 df-reu 2550 . . . 4  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
2 df-rex 2549 . . . . 5  |-  ( E. x  e.  A  (
ph  /\  ps )  <->  E. x ( x  e.  A  /\  ( ph  /\ 
ps ) ) )
3 anass 630 . . . . . 6  |-  ( ( ( x  e.  A  /\  ph )  /\  ps ) 
<->  ( x  e.  A  /\  ( ph  /\  ps ) ) )
43exbii 1569 . . . . 5  |-  ( E. x ( ( x  e.  A  /\  ph )  /\  ps )  <->  E. x
( x  e.  A  /\  ( ph  /\  ps ) ) )
52, 4bitr4i 243 . . . 4  |-  ( E. x  e.  A  (
ph  /\  ps )  <->  E. x ( ( x  e.  A  /\  ph )  /\  ps ) )
6 eupick 2206 . . . 4  |-  ( ( E! x ( x  e.  A  /\  ph )  /\  E. x ( ( x  e.  A  /\  ph )  /\  ps ) )  ->  (
( x  e.  A  /\  ph )  ->  ps ) )
71, 5, 6syl2anb 465 . . 3  |-  ( ( E! x  e.  A  ph 
/\  E. x  e.  A  ( ph  /\  ps )
)  ->  ( (
x  e.  A  /\  ph )  ->  ps )
)
87exp3a 425 . 2  |-  ( ( E! x  e.  A  ph 
/\  E. x  e.  A  ( ph  /\  ps )
)  ->  ( x  e.  A  ->  ( ph  ->  ps ) ) )
983impia 1148 1  |-  ( ( E! x  e.  A  ph 
/\  E. x  e.  A  ( ph  /\  ps )  /\  x  e.  A
)  ->  ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1528    e. wcel 1684   E!weu 2143   E.wrex 2544   E!wreu 2545
This theorem is referenced by:  reupick2  3454
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-rex 2549  df-reu 2550
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