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Theorem sbnfc2 3141
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 2791 . . . . 5  |-  y  e. 
_V
2 csbtt 3093 . . . . 5  |-  ( ( y  e.  _V  /\  F/_ x A )  ->  [_ y  /  x ]_ A  =  A
)
31, 2mpan 651 . . . 4  |-  ( F/_ x A  ->  [_ y  /  x ]_ A  =  A )
4 vex 2791 . . . . 5  |-  z  e. 
_V
5 csbtt 3093 . . . . 5  |-  ( ( z  e.  _V  /\  F/_ x A )  ->  [_ z  /  x ]_ A  =  A
)
64, 5mpan 651 . . . 4  |-  ( F/_ x A  ->  [_ z  /  x ]_ A  =  A )
73, 6eqtr4d 2318 . . 3  |-  ( F/_ x A  ->  [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A )
87alrimivv 1618 . 2  |-  ( F/_ x A  ->  A. y A. z [_ y  /  x ]_ A  =  [_ z  /  x ]_ A
)
9 nfv 1605 . . 3  |-  F/ w A. y A. z [_ y  /  x ]_ A  =  [_ z  /  x ]_ A
10 eleq2 2344 . . . . . 6  |-  ( [_ y  /  x ]_ A  =  [_ z  /  x ]_ A  ->  ( w  e.  [_ y  /  x ]_ A  <->  w  e.  [_ z  /  x ]_ A ) )
11 sbsbc 2995 . . . . . . 7  |-  ( [ y  /  x ]
w  e.  A  <->  [. y  /  x ]. w  e.  A
)
12 sbcel2g 3102 . . . . . . . 8  |-  ( y  e.  _V  ->  ( [. y  /  x ]. w  e.  A  <->  w  e.  [_ y  /  x ]_ A ) )
131, 12ax-mp 8 . . . . . . 7  |-  ( [. y  /  x ]. w  e.  A  <->  w  e.  [_ y  /  x ]_ A )
1411, 13bitri 240 . . . . . 6  |-  ( [ y  /  x ]
w  e.  A  <->  w  e.  [_ y  /  x ]_ A )
15 sbsbc 2995 . . . . . . 7  |-  ( [ z  /  x ]
w  e.  A  <->  [. z  /  x ]. w  e.  A
)
16 sbcel2g 3102 . . . . . . . 8  |-  ( z  e.  _V  ->  ( [. z  /  x ]. w  e.  A  <->  w  e.  [_ z  /  x ]_ A ) )
174, 16ax-mp 8 . . . . . . 7  |-  ( [. z  /  x ]. w  e.  A  <->  w  e.  [_ z  /  x ]_ A )
1815, 17bitri 240 . . . . . 6  |-  ( [ z  /  x ]
w  e.  A  <->  w  e.  [_ z  /  x ]_ A )
1910, 14, 183bitr4g 279 . . . . 5  |-  ( [_ y  /  x ]_ A  =  [_ z  /  x ]_ A  ->  ( [ y  /  x ]
w  e.  A  <->  [ z  /  x ] w  e.  A ) )
20192alimi 1547 . . . 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
) )
21 sbnf2 2047 . . . 4  |-  ( F/ x  w  e.  A  <->  A. y A. z ( [ y  /  x ] w  e.  A  <->  [ z  /  x ]
w  e.  A ) )
2220, 21sylibr 203 . . 3  |-  ( A. y A. z [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A  ->  F/ x  w  e.  A )
239, 22nfcd 2414 . 2  |-  ( A. y A. z [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A  ->  F/_ x A )
248, 23impbii 180 1  |-  ( F/_ x A  <->  A. y A. z [_ y  /  x ]_ A  =  [_ z  /  x ]_ A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1527   F/wnf 1531    = wceq 1623   [wsb 1629    e. wcel 1684   F/_wnfc 2406   _Vcvv 2788   [.wsbc 2991   [_csb 3081
This theorem is referenced by:  eusvnf  4529
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-csb 3082
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