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Theorem eusv2 4533
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 4532 . 2  |-  ( E! y E. x  y  =  A  <->  F/_ x A )
3 eusvnfb 4530 . . 3  |-  ( E! y A. x  y  =  A  <->  ( F/_ x A  /\  A  e. 
_V ) )
41, 3mpbiran2 885 . 2  |-  ( E! y A. x  y  =  A  <->  F/_ x A )
52, 4bitr4i 243 1  |-  ( E! y E. x  y  =  A  <->  E! y A. x  y  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   F/_wnfc 2406   _Vcvv 2788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992  df-csb 3082  df-un 3157  df-sn 3646  df-pr 3647  df-uni 3828
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