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Theorem eusv2 4348
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 4347 . 2  |-  ( E! y E. x  y  =  A  <->  F/_ x A )
3 eusvnfb 4345 . . 3  |-  ( E! y A. x  y  =  A  <->  ( F/_ x A  /\  A  e. 
_V ) )
41, 3mpbiran2 910 . 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 1314    = wceq 1316   E.wex 1453    e. wcel 1465   E!weu 1977   F/_wnfc 2245   _Vcvv 2660
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-sn 3503  df-pr 3504  df-uni 3707
This theorem is referenced by: (None)
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