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Theorem eusv2 4722
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 4721 . 2  |-  ( E! y E. x  y  =  A  <->  F/_ x A )
3 eusvnfb 4719 . . 3  |-  ( E! y A. x  y  =  A  <->  ( F/_ x A  /\  A  e. 
_V ) )
41, 3mpbiran2 886 . 2  |-  ( E! y A. x  y  =  A  <->  F/_ x A )
52, 4bitr4i 244 1  |-  ( E! y E. x  y  =  A  <->  E! y A. x  y  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   E!weu 2281   F/_wnfc 2559   _Vcvv 2956
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-sbc 3162  df-csb 3252  df-un 3325  df-sn 3820  df-pr 3821  df-uni 4016
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