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Theorem reuhyp 4644
Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 4642. (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 1321 . 2  |-  T.
2 reuhyp.1 . . . 4  |-  ( x  e.  C  ->  B  e.  C )
32adantl 452 . . 3  |-  ( (  T.  /\  x  e.  C )  ->  B  e.  C )
4 reuhyp.2 . . . 4  |-  ( ( x  e.  C  /\  y  e.  C )  ->  ( x  =  A  <-> 
y  =  B ) )
543adant1 973 . . 3  |-  ( (  T.  /\  x  e.  C  /\  y  e.  C )  ->  (
x  =  A  <->  y  =  B ) )
63, 5reuhypd 4643 . 2  |-  ( (  T.  /\  x  e.  C )  ->  E! y  e.  C  x  =  A )
71, 6mpan 651 1  |-  ( x  e.  C  ->  E! y  e.  C  x  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    T. wtru 1316    = wceq 1642    e. wcel 1710   E!wreu 2621
This theorem is referenced by:  riotaneg  9819  zmax  10405  rebtwnz  10407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-reu 2626  df-v 2866
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