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Theorem eusvnfb 4252
Description: Two ways to say that  A ( x ) is a set expression that does not depend on  x. (Contributed by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
eusvnfb  |-  ( E! y A. x  y  =  A  <->  ( F/_ x A  /\  A  e. 
_V ) )
Distinct variable groups:    x, y    y, A
Allowed substitution hint:    A( x)

Proof of Theorem eusvnfb
StepHypRef Expression
1 eusvnf 4251 . . 3  |-  ( E! y A. x  y  =  A  ->  F/_ x A )
2 euex 1975 . . . 4  |-  ( E! y A. x  y  =  A  ->  E. y A. x  y  =  A )
3 id 19 . . . . . . 7  |-  ( y  =  A  ->  y  =  A )
4 vex 2618 . . . . . . 7  |-  y  e. 
_V
53, 4syl6eqelr 2176 . . . . . 6  |-  ( y  =  A  ->  A  e.  _V )
65sps 1473 . . . . 5  |-  ( A. x  y  =  A  ->  A  e.  _V )
76exlimiv 1532 . . . 4  |-  ( E. y A. x  y  =  A  ->  A  e.  _V )
82, 7syl 14 . . 3  |-  ( E! y A. x  y  =  A  ->  A  e.  _V )
91, 8jca 300 . 2  |-  ( E! y A. x  y  =  A  ->  ( F/_ x A  /\  A  e.  _V ) )
10 isset 2619 . . . . 5  |-  ( A  e.  _V  <->  E. y 
y  =  A )
11 nfcvd 2226 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x
y )
12 id 19 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x A )
1311, 12nfeqd 2239 . . . . . . 7  |-  ( F/_ x A  ->  F/ x  y  =  A )
1413nfrd 1456 . . . . . 6  |-  ( F/_ x A  ->  ( y  =  A  ->  A. x  y  =  A )
)
1514eximdv 1805 . . . . 5  |-  ( F/_ x A  ->  ( E. y  y  =  A  ->  E. y A. x  y  =  A )
)
1610, 15syl5bi 150 . . . 4  |-  ( F/_ x A  ->  ( A  e.  _V  ->  E. y A. x  y  =  A ) )
1716imp 122 . . 3  |-  ( (
F/_ x A  /\  A  e.  _V )  ->  E. y A. x  y  =  A )
18 eusv1 4250 . . 3  |-  ( E! y A. x  y  =  A  <->  E. y A. x  y  =  A )
1917, 18sylibr 132 . 2  |-  ( (
F/_ x A  /\  A  e.  _V )  ->  E! y A. x  y  =  A )
209, 19impbii 124 1  |-  ( E! y A. x  y  =  A  <->  ( F/_ x A  /\  A  e. 
_V ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103   A.wal 1285    = wceq 1287   E.wex 1424    e. wcel 1436   E!weu 1945   F/_wnfc 2212   _Vcvv 2615
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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-sbc 2830  df-csb 2923
This theorem is referenced by:  eusv2nf  4254  eusv2  4255
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