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Theorem eusvnfb 4475
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 4474 . . 3  |-  ( E! y A. x  y  =  A  ->  F/_ x A )
2 euex 2068 . . . 4  |-  ( E! y A. x  y  =  A  ->  E. y A. x  y  =  A )
3 id 19 . . . . . . 7  |-  ( y  =  A  ->  y  =  A )
4 vex 2755 . . . . . . 7  |-  y  e. 
_V
53, 4eqeltrrdi 2281 . . . . . 6  |-  ( y  =  A  ->  A  e.  _V )
65sps 1548 . . . . 5  |-  ( A. x  y  =  A  ->  A  e.  _V )
76exlimiv 1609 . . . 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 306 . 2  |-  ( E! y A. x  y  =  A  ->  ( F/_ x A  /\  A  e.  _V ) )
10 isset 2758 . . . . 5  |-  ( A  e.  _V  <->  E. y 
y  =  A )
11 nfcvd 2333 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x
y )
12 id 19 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x A )
1311, 12nfeqd 2347 . . . . . . 7  |-  ( F/_ x A  ->  F/ x  y  =  A )
1413nfrd 1531 . . . . . 6  |-  ( F/_ x A  ->  ( y  =  A  ->  A. x  y  =  A )
)
1514eximdv 1891 . . . . 5  |-  ( F/_ x A  ->  ( E. y  y  =  A  ->  E. y A. x  y  =  A )
)
1610, 15biimtrid 152 . . . 4  |-  ( F/_ x A  ->  ( A  e.  _V  ->  E. y A. x  y  =  A ) )
1716imp 124 . . 3  |-  ( (
F/_ x A  /\  A  e.  _V )  ->  E. y A. x  y  =  A )
18 eusv1 4473 . . 3  |-  ( E! y A. x  y  =  A  <->  E. y A. x  y  =  A )
1917, 18sylibr 134 . 2  |-  ( (
F/_ x A  /\  A  e.  _V )  ->  E! y A. x  y  =  A )
209, 19impbii 126 1  |-  ( E! y A. x  y  =  A  <->  ( F/_ x A  /\  A  e. 
_V ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105   A.wal 1362    = wceq 1364   E.wex 1503   E!weu 2038    e. wcel 2160   F/_wnfc 2319   _Vcvv 2752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978  df-csb 3073
This theorem is referenced by:  eusv2nf  4477  eusv2  4478
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