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Theorem eusvnf 4648
Description: Even if  x is free in  A, it is effectively bound when  A ( x ) is single-valued. (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
eusvnf  |-  ( E! y A. x  y  =  A  ->  F/_ x A )
Distinct variable groups:    x, y    y, A
Allowed substitution hint:    A( x)

Proof of Theorem eusvnf
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 euex 2303 . 2  |-  ( E! y A. x  y  =  A  ->  E. y A. x  y  =  A )
2 vex 3121 . . . . . . 7  |-  z  e. 
_V
3 nfcv 2629 . . . . . . . 8  |-  F/_ x
z
4 nfcsb1v 3456 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ A
54nfeq2 2646 . . . . . . . 8  |-  F/ x  y  =  [_ z  /  x ]_ A
6 csbeq1a 3449 . . . . . . . . 9  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
76eqeq2d 2481 . . . . . . . 8  |-  ( x  =  z  ->  (
y  =  A  <->  y  =  [_ z  /  x ]_ A ) )
83, 5, 7spcgf 3198 . . . . . . 7  |-  ( z  e.  _V  ->  ( A. x  y  =  A  ->  y  =  [_ z  /  x ]_ A
) )
92, 8ax-mp 5 . . . . . 6  |-  ( A. x  y  =  A  ->  y  =  [_ z  /  x ]_ A )
10 vex 3121 . . . . . . 7  |-  w  e. 
_V
11 nfcv 2629 . . . . . . . 8  |-  F/_ x w
12 nfcsb1v 3456 . . . . . . . . 9  |-  F/_ x [_ w  /  x ]_ A
1312nfeq2 2646 . . . . . . . 8  |-  F/ x  y  =  [_ w  /  x ]_ A
14 csbeq1a 3449 . . . . . . . . 9  |-  ( x  =  w  ->  A  =  [_ w  /  x ]_ A )
1514eqeq2d 2481 . . . . . . . 8  |-  ( x  =  w  ->  (
y  =  A  <->  y  =  [_ w  /  x ]_ A ) )
1611, 13, 15spcgf 3198 . . . . . . 7  |-  ( w  e.  _V  ->  ( A. x  y  =  A  ->  y  =  [_ w  /  x ]_ A
) )
1710, 16ax-mp 5 . . . . . 6  |-  ( A. x  y  =  A  ->  y  =  [_ w  /  x ]_ A )
189, 17eqtr3d 2510 . . . . 5  |-  ( A. x  y  =  A  ->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
1918alrimivv 1696 . . . 4  |-  ( A. x  y  =  A  ->  A. z A. w [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
20 sbnfc2 3859 . . . 4  |-  ( F/_ x A  <->  A. z A. w [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
2119, 20sylibr 212 . . 3  |-  ( A. x  y  =  A  -> 
F/_ x A )
2221exlimiv 1698 . 2  |-  ( E. y A. x  y  =  A  ->  F/_ x A )
231, 22syl 16 1  |-  ( E! y A. x  y  =  A  ->  F/_ x A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767   E!weu 2275   F/_wnfc 2615   _Vcvv 3118   [_csb 3440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-in 3488  df-ss 3495  df-nul 3791
This theorem is referenced by:  eusvnfb  4649  eusv2i  4650
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