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Theorem eusvnfb 4563
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 4562 . . 3  |-  ( E! y A. x  y  =  A  ->  F/_ x A )
2 euex 2300 . . . 4  |-  ( E! y A. x  y  =  A  ->  E. y A. x  y  =  A )
3 eqvisset 3030 . . . . . 6  |-  ( y  =  A  ->  A  e.  _V )
43sps 1920 . . . . 5  |-  ( A. x  y  =  A  ->  A  e.  _V )
54exlimiv 1770 . . . 4  |-  ( E. y A. x  y  =  A  ->  A  e.  _V )
62, 5syl 17 . . 3  |-  ( E! y A. x  y  =  A  ->  A  e.  _V )
71, 6jca 534 . 2  |-  ( E! y A. x  y  =  A  ->  ( F/_ x A  /\  A  e.  _V ) )
8 isset 3026 . . . . 5  |-  ( A  e.  _V  <->  E. y 
y  =  A )
9 nfcvd 2570 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x
y )
10 id 22 . . . . . . . 8  |-  ( F/_ x A  ->  F/_ x A )
119, 10nfeqd 2576 . . . . . . 7  |-  ( F/_ x A  ->  F/ x  y  =  A )
1211nfrd 1930 . . . . . 6  |-  ( F/_ x A  ->  ( y  =  A  ->  A. x  y  =  A )
)
1312eximdv 1758 . . . . 5  |-  ( F/_ x A  ->  ( E. y  y  =  A  ->  E. y A. x  y  =  A )
)
148, 13syl5bi 220 . . . 4  |-  ( F/_ x A  ->  ( A  e.  _V  ->  E. y A. x  y  =  A ) )
1514imp 430 . . 3  |-  ( (
F/_ x A  /\  A  e.  _V )  ->  E. y A. x  y  =  A )
16 eusv1 4561 . . 3  |-  ( E! y A. x  y  =  A  <->  E. y A. x  y  =  A )
1715, 16sylibr 215 . 2  |-  ( (
F/_ x A  /\  A  e.  _V )  ->  E! y A. x  y  =  A )
187, 17impbii 190 1  |-  ( E! y A. x  y  =  A  <->  ( F/_ x A  /\  A  e. 
_V ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   E.wex 1657    e. wcel 1872   E!weu 2276   F/_wnfc 2556   _Vcvv 3022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-in 3386  df-ss 3393  df-nul 3705
This theorem is referenced by:  eusv2nf  4565  eusv2  4566
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