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Theorem reuhyp 4641
Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 4639. (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 1458 . 2  |- T.
2 reuhyp.1 . . . 4  |-  ( x  e.  C  ->  B  e.  C )
32adantl 472 . . 3  |-  ( ( T.  /\  x  e.  C )  ->  B  e.  C )
4 reuhyp.2 . . . 4  |-  ( ( x  e.  C  /\  y  e.  C )  ->  ( x  =  A  <-> 
y  =  B ) )
543adant1 1032 . . 3  |-  ( ( T.  /\  x  e.  C  /\  y  e.  C )  ->  (
x  =  A  <->  y  =  B ) )
63, 5reuhypd 4640 . 2  |-  ( ( T.  /\  x  e.  C )  ->  E! y  e.  C  x  =  A )
71, 6mpan 681 1  |-  ( x  e.  C  ->  E! y  e.  C  x  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1454   T. wtru 1455    e. wcel 1897   E!wreu 2750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-reu 2755  df-v 3058
This theorem is referenced by:  riotaneg  10613  zriotaneg  11077  zmax  11289  rebtwnz  11291
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