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Theorem reuhypd 4644
Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 6293. (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 3090 . . . . 5  |-  ( B  e.  C  ->  B  e.  _V )
31, 2syl 17 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  B  e.  _V )
4 eueq 3243 . . . 4  |-  ( B  e.  _V  <->  E! y 
y  =  B )
53, 4sylib 199 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  E! y  y  =  B
)
6 eleq1 2494 . . . . . . 7  |-  ( y  =  B  ->  (
y  e.  C  <->  B  e.  C ) )
71, 6syl5ibrcom 225 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  (
y  =  B  -> 
y  e.  C ) )
87pm4.71rd 639 . . . . 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 1205 . . . . . 6  |-  ( ( ( ph  /\  x  e.  C )  /\  y  e.  C )  ->  (
x  =  A  <->  y  =  B ) )
1110pm5.32da 645 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  (
( y  e.  C  /\  x  =  A
)  <->  ( y  e.  C  /\  y  =  B ) ) )
128, 11bitr4d 259 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  (
y  =  B  <->  ( y  e.  C  /\  x  =  A ) ) )
1312eubidv 2286 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  ( E! y  y  =  B 
<->  E! y ( y  e.  C  /\  x  =  A ) ) )
145, 13mpbid 213 . 2  |-  ( (
ph  /\  x  e.  C )  ->  E! y ( y  e.  C  /\  x  =  A ) )
15 df-reu 2782 . 2  |-  ( E! y  e.  C  x  =  A  <->  E! y
( y  e.  C  /\  x  =  A
) )
1614, 15sylibr 215 1  |-  ( (
ph  /\  x  e.  C )  ->  E! y  e.  C  x  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   E!weu 2265   E!wreu 2777   _Vcvv 3081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-reu 2782  df-v 3083
This theorem is referenced by:  reuhyp  4645  riotaocN  32693
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