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Theorem reuhyp 4620
Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 4618. (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 1374 . 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 1006 . . 3  |-  ( ( T.  /\  x  e.  C  /\  y  e.  C )  ->  (
x  =  A  <->  y  =  B ) )
63, 5reuhypd 4619 . 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 1370   T. wtru 1371    e. wcel 1758   E!wreu 2797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-reu 2802  df-v 3072
This theorem is referenced by:  riotaneg  10408  zriotaneg  10857  zmax  11053  rebtwnz  11055
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