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Theorem reuhyp 4668
Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 4666. (Contributed by NM, 15-Nov-2004.)
Hypotheses
Ref Expression
reuhyp.1  |-  ( x  e.  C  ->  B  e.  C )
reuhyp.2  |-  ( ( x  e.  C  /\  y  e.  C )  ->  ( x  =  A  <-> 
y  =  B ) )
Assertion
Ref Expression
reuhyp  |-  ( x  e.  C  ->  E! y  e.  C  x  =  A )
Distinct variable groups:    y, B    y, C    x, y
Allowed substitution hints:    A( x, y)    B( x)    C( x)

Proof of Theorem reuhyp
StepHypRef Expression
1 tru 1378 . 2  |- T.
2 reuhyp.1 . . . 4  |-  ( x  e.  C  ->  B  e.  C )
32adantl 466 . . 3  |-  ( ( T.  /\  x  e.  C )  ->  B  e.  C )
4 reuhyp.2 . . . 4  |-  ( ( x  e.  C  /\  y  e.  C )  ->  ( x  =  A  <-> 
y  =  B ) )
543adant1 1009 . . 3  |-  ( ( T.  /\  x  e.  C  /\  y  e.  C )  ->  (
x  =  A  <->  y  =  B ) )
63, 5reuhypd 4667 . 2  |-  ( ( T.  /\  x  e.  C )  ->  E! y  e.  C  x  =  A )
71, 6mpan 670 1  |-  ( x  e.  C  ->  E! y  e.  C  x  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374   T. wtru 1375    e. wcel 1762   E!wreu 2809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-reu 2814  df-v 3108
This theorem is referenced by:  riotaneg  10507  zriotaneg  10963  zmax  11168  rebtwnz  11170
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