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Theorem reuhypd 4752
Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 6583. (Contributed by NM, 16-Jan-2012.)
Hypotheses
Ref Expression
reuhypd.1  |-  ( (
ph  /\  x  e.  C )  ->  B  e.  C )
reuhypd.2  |-  ( (
ph  /\  x  e.  C  /\  y  e.  C
)  ->  ( x  =  A  <->  y  =  B ) )
Assertion
Ref Expression
reuhypd  |-  ( (
ph  /\  x  e.  C )  ->  E! y  e.  C  x  =  A )
Distinct variable groups:    ph, y    y, B    y, C    x, y
Allowed substitution hints:    ph( x)    A( x, y)    B( x)    C( x)

Proof of Theorem reuhypd
StepHypRef Expression
1 reuhypd.1 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  B  e.  C )
2 elex 2966 . . . . 5  |-  ( B  e.  C  ->  B  e.  _V )
31, 2syl 16 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  B  e.  _V )
4 eueq 3108 . . . 4  |-  ( B  e.  _V  <->  E! y 
y  =  B )
53, 4sylib 190 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  E! y  y  =  B
)
6 eleq1 2498 . . . . . . 7  |-  ( y  =  B  ->  (
y  e.  C  <->  B  e.  C ) )
71, 6syl5ibrcom 215 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  (
y  =  B  -> 
y  e.  C ) )
87pm4.71rd 618 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  (
y  =  B  <->  ( y  e.  C  /\  y  =  B ) ) )
9 reuhypd.2 . . . . . . 7  |-  ( (
ph  /\  x  e.  C  /\  y  e.  C
)  ->  ( x  =  A  <->  y  =  B ) )
1093expa 1154 . . . . . 6  |-  ( ( ( ph  /\  x  e.  C )  /\  y  e.  C )  ->  (
x  =  A  <->  y  =  B ) )
1110pm5.32da 624 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  (
( y  e.  C  /\  x  =  A
)  <->  ( y  e.  C  /\  y  =  B ) ) )
128, 11bitr4d 249 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  (
y  =  B  <->  ( y  e.  C  /\  x  =  A ) ) )
1312eubidv 2291 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  ( E! y  y  =  B 
<->  E! y ( y  e.  C  /\  x  =  A ) ) )
145, 13mpbid 203 . 2  |-  ( (
ph  /\  x  e.  C )  ->  E! y ( y  e.  C  /\  x  =  A ) )
15 df-reu 2714 . 2  |-  ( E! y  e.  C  x  =  A  <->  E! y
( y  e.  C  /\  x  =  A
) )
1614, 15sylibr 205 1  |-  ( (
ph  /\  x  e.  C )  ->  E! y  e.  C  x  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E!weu 2283   E!wreu 2709   _Vcvv 2958
This theorem is referenced by:  reuhyp  4753  riotaocN  30069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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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