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Theorem eusv4 4672
Description: Two ways to express single-valuedness of a class expression  B ( x ). (Contributed by NM, 27-Oct-2010.)
Hypothesis
Ref Expression
eusv4.1  |-  B  e. 
_V
Assertion
Ref Expression
eusv4  |-  ( E! x E. y  e.  A  x  =  B  <-> 
E! x A. y  e.  A  x  =  B )
Distinct variable groups:    x, y, A    x, B
Allowed substitution hint:    B( y)

Proof of Theorem eusv4
StepHypRef Expression
1 reusv2lem3 4666 . 2  |-  ( A. y  e.  A  B  e.  _V  ->  ( E! x E. y  e.  A  x  =  B  <->  E! x A. y  e.  A  x  =  B )
)
2 eusv4.1 . . 3  |-  B  e. 
_V
32a1i 11 . 2  |-  ( y  e.  A  ->  B  e.  _V )
41, 3mprg 2718 1  |-  ( E! x E. y  e.  A  x  =  B  <-> 
E! x A. y  e.  A  x  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1717   E!weu 2238   A.wral 2649   E.wrex 2650   _Vcvv 2899
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-nul 4279  ax-pow 4318
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-v 2901  df-dif 3266  df-nul 3572
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