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Theorem eusvnfb 4186
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 4185 . . 3  |-  ( E! y A. x  y  =  A  ->  F/_ x A )
2 euex 1930 . . . 4  |-  ( E! y A. x  y  =  A  ->  E. y A. x  y  =  A )
3 id 19 . . . . . . 7  |-  ( y  =  A  ->  y  =  A )
4 vex 2560 . . . . . . 7  |-  y  e. 
_V
53, 4syl6eqelr 2129 . . . . . 6  |-  ( y  =  A  ->  A  e.  _V )
65sps 1430 . . . . 5  |-  ( A. x  y  =  A  ->  A  e.  _V )
76exlimiv 1489 . . . 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 290 . 2  |-  ( E! y A. x  y  =  A  ->  ( F/_ x A  /\  A  e.  _V ) )
10 isset 2561 . . . . 5  |-  ( A  e.  _V  <->  E. y 
y  =  A )
11 nfcvd 2179 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x
y )
12 id 19 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x A )
1311, 12nfeqd 2192 . . . . . . 7  |-  ( F/_ x A  ->  F/ x  y  =  A )
1413nfrd 1413 . . . . . 6  |-  ( F/_ x A  ->  ( y  =  A  ->  A. x  y  =  A )
)
1514eximdv 1760 . . . . 5  |-  ( F/_ x A  ->  ( E. y  y  =  A  ->  E. y A. x  y  =  A )
)
1610, 15syl5bi 141 . . . 4  |-  ( F/_ x A  ->  ( A  e.  _V  ->  E. y A. x  y  =  A ) )
1716imp 115 . . 3  |-  ( (
F/_ x A  /\  A  e.  _V )  ->  E. y A. x  y  =  A )
18 eusv1 4184 . . 3  |-  ( E! y A. x  y  =  A  <->  E. y A. x  y  =  A )
1917, 18sylibr 137 . 2  |-  ( (
F/_ x A  /\  A  e.  _V )  ->  E! y A. x  y  =  A )
209, 19impbii 117 1  |-  ( E! y A. x  y  =  A  <->  ( F/_ x A  /\  A  e. 
_V ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 97    <-> wb 98   A.wal 1241    = wceq 1243   E.wex 1381    e. wcel 1393   E!weu 1900   F/_wnfc 2165   _Vcvv 2557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853
This theorem is referenced by:  eusv2nf  4188  eusv2  4189
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