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Theorem eusv2 4373
Description: Two ways to express single-valuedness of a class expression  A ( x ). (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
Hypothesis
Ref Expression
eusv2.1  |-  A  e. 
_V
Assertion
Ref Expression
eusv2  |-  ( E! y E. x  y  =  A  <->  E! y A. x  y  =  A )
Distinct variable groups:    x, y    y, A
Allowed substitution hint:    A( x)

Proof of Theorem eusv2
StepHypRef Expression
1 eusv2.1 . . 3  |-  A  e. 
_V
21eusv2nf 4372 . 2  |-  ( E! y E. x  y  =  A  <->  F/_ x A )
3 eusvnfb 4370 . . 3  |-  ( E! y A. x  y  =  A  <->  ( F/_ x A  /\  A  e. 
_V ) )
41, 3mpbiran2 925 . 2  |-  ( E! y A. x  y  =  A  <->  F/_ x A )
52, 4bitr4i 186 1  |-  ( E! y E. x  y  =  A  <->  E! y A. x  y  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1329    = wceq 1331   E.wex 1468    e. wcel 1480   E!weu 1997   F/_wnfc 2266   _Vcvv 2681
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-sn 3528  df-pr 3529  df-uni 3732
This theorem is referenced by: (None)
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