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Theorem eusvnfb 4567
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 4566 . . 3  |-  ( E! y A. x  y  =  A  ->  F/_ x A )
2 euex 2199 . . . 4  |-  ( E! y A. x  y  =  A  ->  E. y A. x  y  =  A )
3 id 19 . . . . . . 7  |-  ( y  =  A  ->  y  =  A )
4 vex 2825 . . . . . . 7  |-  y  e. 
_V
53, 4syl6eqelr 2405 . . . . . 6  |-  ( y  =  A  ->  A  e.  _V )
65sps 1736 . . . . 5  |-  ( A. x  y  =  A  ->  A  e.  _V )
76exlimiv 1625 . . . 4  |-  ( E. y A. x  y  =  A  ->  A  e.  _V )
82, 7syl 15 . . 3  |-  ( E! y A. x  y  =  A  ->  A  e.  _V )
91, 8jca 518 . 2  |-  ( E! y A. x  y  =  A  ->  ( F/_ x A  /\  A  e.  _V ) )
10 isset 2826 . . . . 5  |-  ( A  e.  _V  <->  E. y 
y  =  A )
11 nfcvd 2453 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x
y )
12 id 19 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x A )
1311, 12nfeqd 2466 . . . . . . 7  |-  ( F/_ x A  ->  F/ x  y  =  A )
1413nfrd 1767 . . . . . 6  |-  ( F/_ x A  ->  ( y  =  A  ->  A. x  y  =  A )
)
1514eximdv 1613 . . . . 5  |-  ( F/_ x A  ->  ( E. y  y  =  A  ->  E. y A. x  y  =  A )
)
1610, 15syl5bi 208 . . . 4  |-  ( F/_ x A  ->  ( A  e.  _V  ->  E. y A. x  y  =  A ) )
1716imp 418 . . 3  |-  ( (
F/_ x A  /\  A  e.  _V )  ->  E. y A. x  y  =  A )
18 eusv1 4565 . . 3  |-  ( E! y A. x  y  =  A  <->  E. y A. x  y  =  A )
1917, 18sylibr 203 . 2  |-  ( (
F/_ x A  /\  A  e.  _V )  ->  E! y A. x  y  =  A )
209, 19impbii 180 1  |-  ( E! y A. x  y  =  A  <->  ( F/_ x A  /\  A  e. 
_V ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   A.wal 1531   E.wex 1532    = wceq 1633    e. wcel 1701   E!weu 2176   F/_wnfc 2439   _Vcvv 2822
This theorem is referenced by:  eusv2nf  4569  eusv2  4570
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-sbc 3026  df-csb 3116
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