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Theorem eusv2 4270
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 4269 . 2  |-  ( E! y E. x  y  =  A  <->  F/_ x A )
3 eusvnfb 4267 . . 3  |-  ( E! y A. x  y  =  A  <->  ( F/_ x A  /\  A  e. 
_V ) )
41, 3mpbiran2 887 . 2  |-  ( E! y A. x  y  =  A  <->  F/_ x A )
52, 4bitr4i 185 1  |-  ( E! y E. x  y  =  A  <->  E! y A. x  y  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   A.wal 1287    = wceq 1289   E.wex 1426    e. wcel 1438   E!weu 1948   F/_wnfc 2215   _Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-sn 3447  df-pr 3448  df-uni 3649
This theorem is referenced by: (None)
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