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Theorem eusv4 4724
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 4718 . 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 2767 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 1652    e. wcel 1725   E!weu 2280   A.wral 2697   E.wrex 2698   _Vcvv 2948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330  ax-pow 4369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-nul 3621
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