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Theorem reuhyp 4754
Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 4752. (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 1331 . 2  |-  T.
2 reuhyp.1 . . . 4  |-  ( x  e.  C  ->  B  e.  C )
32adantl 454 . . 3  |-  ( (  T.  /\  x  e.  C )  ->  B  e.  C )
4 reuhyp.2 . . . 4  |-  ( ( x  e.  C  /\  y  e.  C )  ->  ( x  =  A  <-> 
y  =  B ) )
543adant1 976 . . 3  |-  ( (  T.  /\  x  e.  C  /\  y  e.  C )  ->  (
x  =  A  <->  y  =  B ) )
63, 5reuhypd 4753 . 2  |-  ( (  T.  /\  x  e.  C )  ->  E! y  e.  C  x  =  A )
71, 6mpan 653 1  |-  ( x  e.  C  ->  E! y  e.  C  x  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    T. wtru 1326    = wceq 1653    e. wcel 1726   E!wreu 2709
This theorem is referenced by:  riotaneg  9988  zmax  10576  rebtwnz  10578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-reu 2714  df-v 2960
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