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Theorem sbnfc2 3815
Description: Two ways of expressing " x is (effectively) not free in  A." (Contributed by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
sbnfc2  |-  ( F/_ x A  <->  A. y A. z [_ y  /  x ]_ A  =  [_ z  /  x ]_ A )
Distinct variable groups:    x, y,
z    y, A, z
Allowed substitution hint:    A( x)

Proof of Theorem sbnfc2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 vex 3081 . . . . 5  |-  y  e. 
_V
2 csbtt 3407 . . . . 5  |-  ( ( y  e.  _V  /\  F/_ x A )  ->  [_ y  /  x ]_ A  =  A
)
31, 2mpan 670 . . . 4  |-  ( F/_ x A  ->  [_ y  /  x ]_ A  =  A )
4 vex 3081 . . . . 5  |-  z  e. 
_V
5 csbtt 3407 . . . . 5  |-  ( ( z  e.  _V  /\  F/_ x A )  ->  [_ z  /  x ]_ A  =  A
)
64, 5mpan 670 . . . 4  |-  ( F/_ x A  ->  [_ z  /  x ]_ A  =  A )
73, 6eqtr4d 2498 . . 3  |-  ( F/_ x A  ->  [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A )
87alrimivv 1687 . 2  |-  ( F/_ x A  ->  A. y A. z [_ y  /  x ]_ A  =  [_ z  /  x ]_ A
)
9 nfv 1674 . . 3  |-  F/ w A. y A. z [_ y  /  x ]_ A  =  [_ z  /  x ]_ A
10 eleq2 2527 . . . . . 6  |-  ( [_ y  /  x ]_ A  =  [_ z  /  x ]_ A  ->  ( w  e.  [_ y  /  x ]_ A  <->  w  e.  [_ z  /  x ]_ A ) )
11 sbsbc 3298 . . . . . . 7  |-  ( [ y  /  x ]
w  e.  A  <->  [. y  /  x ]. w  e.  A
)
12 sbcel2 3792 . . . . . . 7  |-  ( [. y  /  x ]. w  e.  A  <->  w  e.  [_ y  /  x ]_ A )
1311, 12bitri 249 . . . . . 6  |-  ( [ y  /  x ]
w  e.  A  <->  w  e.  [_ y  /  x ]_ A )
14 sbsbc 3298 . . . . . . 7  |-  ( [ z  /  x ]
w  e.  A  <->  [. z  /  x ]. w  e.  A
)
15 sbcel2 3792 . . . . . . 7  |-  ( [. z  /  x ]. w  e.  A  <->  w  e.  [_ z  /  x ]_ A )
1614, 15bitri 249 . . . . . 6  |-  ( [ z  /  x ]
w  e.  A  <->  w  e.  [_ z  /  x ]_ A )
1710, 13, 163bitr4g 288 . . . . 5  |-  ( [_ y  /  x ]_ A  =  [_ z  /  x ]_ A  ->  ( [ y  /  x ]
w  e.  A  <->  [ z  /  x ] w  e.  A ) )
18172alimi 1606 . . . 4  |-  ( A. y A. z [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A  ->  A. y A. z ( [ y  /  x ] w  e.  A  <->  [ z  /  x ] w  e.  A
) )
19 sbnf2 2153 . . . 4  |-  ( F/ x  w  e.  A  <->  A. y A. z ( [ y  /  x ] w  e.  A  <->  [ z  /  x ]
w  e.  A ) )
2018, 19sylibr 212 . . 3  |-  ( A. y A. z [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A  ->  F/ x  w  e.  A )
219, 20nfcd 2610 . 2  |-  ( A. y A. z [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A  ->  F/_ x A )
228, 21impbii 188 1  |-  ( F/_ x A  <->  A. y A. z [_ y  /  x ]_ A  =  [_ z  /  x ]_ A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wal 1368    = wceq 1370   F/wnf 1590   [wsb 1702    e. wcel 1758   F/_wnfc 2602   _Vcvv 3078   [.wsbc 3294   [_csb 3396
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-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3747
This theorem is referenced by:  eusvnf  4596
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