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Theorem eusvnfb 4561
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 4560 . . 3  |-  ( E! y A. x  y  =  A  ->  F/_ x A )
2 euex 2244 . . . 4  |-  ( E! y A. x  y  =  A  ->  E. y A. x  y  =  A )
3 eqvisset 3042 . . . . . 6  |-  ( y  =  A  ->  A  e.  _V )
43sps 1873 . . . . 5  |-  ( A. x  y  =  A  ->  A  e.  _V )
54exlimiv 1730 . . . 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 530 . 2  |-  ( E! y A. x  y  =  A  ->  ( F/_ x A  /\  A  e.  _V ) )
8 isset 3038 . . . . 5  |-  ( A  e.  _V  <->  E. y 
y  =  A )
9 nfcvd 2545 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x
y )
10 id 22 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x A )
119, 10nfeqd 2551 . . . . . . 7  |-  ( F/_ x A  ->  F/ x  y  =  A )
1211nfrd 1883 . . . . . 6  |-  ( F/_ x A  ->  ( y  =  A  ->  A. x  y  =  A )
)
1312eximdv 1718 . . . . 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 427 . . 3  |-  ( (
F/_ x A  /\  A  e.  _V )  ->  E. y A. x  y  =  A )
16 eusv1 4559 . . 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 367   A.wal 1397    = wceq 1399   E.wex 1620    e. wcel 1826   E!weu 2218   F/_wnfc 2530   _Vcvv 3034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-in 3396  df-ss 3403  df-nul 3712
This theorem is referenced by:  eusv2nf  4563  eusv2  4564
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