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Theorem eusvnfb 4595
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 4594 . . 3  |-  ( E! y A. x  y  =  A  ->  F/_ x A )
2 euex 2290 . . . 4  |-  ( E! y A. x  y  =  A  ->  E. y A. x  y  =  A )
3 eqvisset 3084 . . . . . 6  |-  ( y  =  A  ->  A  e.  _V )
43sps 1805 . . . . 5  |-  ( A. x  y  =  A  ->  A  e.  _V )
54exlimiv 1689 . . . 4  |-  ( E. y A. x  y  =  A  ->  A  e.  _V )
62, 5syl 16 . . 3  |-  ( E! y A. x  y  =  A  ->  A  e.  _V )
71, 6jca 532 . 2  |-  ( E! y A. x  y  =  A  ->  ( F/_ x A  /\  A  e.  _V ) )
8 isset 3080 . . . . 5  |-  ( A  e.  _V  <->  E. y 
y  =  A )
9 nfcvd 2617 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x
y )
10 id 22 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x A )
119, 10nfeqd 2623 . . . . . . 7  |-  ( F/_ x A  ->  F/ x  y  =  A )
1211nfrd 1814 . . . . . 6  |-  ( F/_ x A  ->  ( y  =  A  ->  A. x  y  =  A )
)
1312eximdv 1677 . . . . 5  |-  ( F/_ x A  ->  ( E. y  y  =  A  ->  E. y A. x  y  =  A )
)
148, 13syl5bi 217 . . . 4  |-  ( F/_ x A  ->  ( A  e.  _V  ->  E. y A. x  y  =  A ) )
1514imp 429 . . 3  |-  ( (
F/_ x A  /\  A  e.  _V )  ->  E. y A. x  y  =  A )
16 eusv1 4593 . . 3  |-  ( E! y A. x  y  =  A  <->  E. y A. x  y  =  A )
1715, 16sylibr 212 . 2  |-  ( (
F/_ x A  /\  A  e.  _V )  ->  E! y A. x  y  =  A )
187, 17impbii 188 1  |-  ( E! y A. x  y  =  A  <->  ( F/_ x A  /\  A  e. 
_V ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   A.wal 1368    = wceq 1370   E.wex 1587    e. wcel 1758   E!weu 2262   F/_wnfc 2602   _Vcvv 3076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-in 3442  df-ss 3449  df-nul 3745
This theorem is referenced by:  eusv2nf  4597  eusv2  4598
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