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Theorem sbnfc2 3796
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 3048 . . . . 5  |-  y  e. 
_V
2 csbtt 3374 . . . . 5  |-  ( ( y  e.  _V  /\  F/_ x A )  ->  [_ y  /  x ]_ A  =  A
)
31, 2mpan 676 . . . 4  |-  ( F/_ x A  ->  [_ y  /  x ]_ A  =  A )
4 vex 3048 . . . . 5  |-  z  e. 
_V
5 csbtt 3374 . . . . 5  |-  ( ( z  e.  _V  /\  F/_ x A )  ->  [_ z  /  x ]_ A  =  A
)
64, 5mpan 676 . . . 4  |-  ( F/_ x A  ->  [_ z  /  x ]_ A  =  A )
73, 6eqtr4d 2488 . . 3  |-  ( F/_ x A  ->  [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A )
87alrimivv 1774 . 2  |-  ( F/_ x A  ->  A. y A. z [_ y  /  x ]_ A  =  [_ z  /  x ]_ A
)
9 nfv 1761 . . 3  |-  F/ w A. y A. z [_ y  /  x ]_ A  =  [_ z  /  x ]_ A
10 eleq2 2518 . . . . . 6  |-  ( [_ y  /  x ]_ A  =  [_ z  /  x ]_ A  ->  ( w  e.  [_ y  /  x ]_ A  <->  w  e.  [_ z  /  x ]_ A ) )
11 sbsbc 3271 . . . . . . 7  |-  ( [ y  /  x ]
w  e.  A  <->  [. y  /  x ]. w  e.  A
)
12 sbcel2 3778 . . . . . . 7  |-  ( [. y  /  x ]. w  e.  A  <->  w  e.  [_ y  /  x ]_ A )
1311, 12bitri 253 . . . . . 6  |-  ( [ y  /  x ]
w  e.  A  <->  w  e.  [_ y  /  x ]_ A )
14 sbsbc 3271 . . . . . . 7  |-  ( [ z  /  x ]
w  e.  A  <->  [. z  /  x ]. w  e.  A
)
15 sbcel2 3778 . . . . . . 7  |-  ( [. z  /  x ]. w  e.  A  <->  w  e.  [_ z  /  x ]_ A )
1614, 15bitri 253 . . . . . 6  |-  ( [ z  /  x ]
w  e.  A  <->  w  e.  [_ z  /  x ]_ A )
1710, 13, 163bitr4g 292 . . . . 5  |-  ( [_ y  /  x ]_ A  =  [_ z  /  x ]_ A  ->  ( [ y  /  x ]
w  e.  A  <->  [ z  /  x ] w  e.  A ) )
18172alimi 1685 . . . 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 2268 . . . 4  |-  ( F/ x  w  e.  A  <->  A. y A. z ( [ y  /  x ] w  e.  A  <->  [ z  /  x ]
w  e.  A ) )
2018, 19sylibr 216 . . 3  |-  ( A. y A. z [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A  ->  F/ x  w  e.  A )
219, 20nfcd 2587 . 2  |-  ( A. y A. z [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A  ->  F/_ x A )
228, 21impbii 191 1  |-  ( F/_ x A  <->  A. y A. z [_ y  /  x ]_ A  =  [_ z  /  x ]_ A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188   A.wal 1442    = wceq 1444   F/wnf 1667   [wsb 1797    e. wcel 1887   F/_wnfc 2579   _Vcvv 3045   [.wsbc 3267   [_csb 3363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-in 3411  df-ss 3418  df-nul 3732
This theorem is referenced by:  eusvnf  4598
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