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Theorem reuhypd 4456
Description: A theorem useful for eliminating restricted existential uniqueness hypotheses. (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 2741 . . . . 5  |-  ( B  e.  C  ->  B  e.  _V )
31, 2syl 14 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  B  e.  _V )
4 eueq 2901 . . . 4  |-  ( B  e.  _V  <->  E! y 
y  =  B )
53, 4sylib 121 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  E! y  y  =  B
)
6 eleq1 2233 . . . . . . 7  |-  ( y  =  B  ->  (
y  e.  C  <->  B  e.  C ) )
71, 6syl5ibrcom 156 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  (
y  =  B  -> 
y  e.  C ) )
87pm4.71rd 392 . . . . 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 1198 . . . . . 6  |-  ( ( ( ph  /\  x  e.  C )  /\  y  e.  C )  ->  (
x  =  A  <->  y  =  B ) )
1110pm5.32da 449 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  (
( y  e.  C  /\  x  =  A
)  <->  ( y  e.  C  /\  y  =  B ) ) )
128, 11bitr4d 190 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  (
y  =  B  <->  ( y  e.  C  /\  x  =  A ) ) )
1312eubidv 2027 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  ( E! y  y  =  B 
<->  E! y ( y  e.  C  /\  x  =  A ) ) )
145, 13mpbid 146 . 2  |-  ( (
ph  /\  x  e.  C )  ->  E! y ( y  e.  C  /\  x  =  A ) )
15 df-reu 2455 . 2  |-  ( E! y  e.  C  x  =  A  <->  E! y
( y  e.  C  /\  x  =  A
) )
1614, 15sylibr 133 1  |-  ( (
ph  /\  x  e.  C )  ->  E! y  e.  C  x  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 973    = wceq 1348   E!weu 2019    e. wcel 2141   E!wreu 2450   _Vcvv 2730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-reu 2455  df-v 2732
This theorem is referenced by:  reuhyp  4457
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