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Theorem sbnfc2 3055
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 2684 . . . . 5  |-  y  e. 
_V
2 csbtt 3009 . . . . 5  |-  ( ( y  e.  _V  /\  F/_ x A )  ->  [_ y  /  x ]_ A  =  A
)
31, 2mpan 420 . . . 4  |-  ( F/_ x A  ->  [_ y  /  x ]_ A  =  A )
4 vex 2684 . . . . 5  |-  z  e. 
_V
5 csbtt 3009 . . . . 5  |-  ( ( z  e.  _V  /\  F/_ x A )  ->  [_ z  /  x ]_ A  =  A
)
64, 5mpan 420 . . . 4  |-  ( F/_ x A  ->  [_ z  /  x ]_ A  =  A )
73, 6eqtr4d 2173 . . 3  |-  ( F/_ x A  ->  [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A )
87alrimivv 1847 . 2  |-  ( F/_ x A  ->  A. y A. z [_ y  /  x ]_ A  =  [_ z  /  x ]_ A
)
9 nfv 1508 . . 3  |-  F/ w A. y A. z [_ y  /  x ]_ A  =  [_ z  /  x ]_ A
10 eleq2 2201 . . . . . 6  |-  ( [_ y  /  x ]_ A  =  [_ z  /  x ]_ A  ->  ( w  e.  [_ y  /  x ]_ A  <->  w  e.  [_ z  /  x ]_ A ) )
11 sbsbc 2908 . . . . . . 7  |-  ( [ y  /  x ]
w  e.  A  <->  [. y  /  x ]. w  e.  A
)
12 sbcel2g 3018 . . . . . . . 8  |-  ( y  e.  _V  ->  ( [. y  /  x ]. w  e.  A  <->  w  e.  [_ y  /  x ]_ A ) )
131, 12ax-mp 5 . . . . . . 7  |-  ( [. y  /  x ]. w  e.  A  <->  w  e.  [_ y  /  x ]_ A )
1411, 13bitri 183 . . . . . 6  |-  ( [ y  /  x ]
w  e.  A  <->  w  e.  [_ y  /  x ]_ A )
15 sbsbc 2908 . . . . . . 7  |-  ( [ z  /  x ]
w  e.  A  <->  [. z  /  x ]. w  e.  A
)
16 sbcel2g 3018 . . . . . . . 8  |-  ( z  e.  _V  ->  ( [. z  /  x ]. w  e.  A  <->  w  e.  [_ z  /  x ]_ A ) )
174, 16ax-mp 5 . . . . . . 7  |-  ( [. z  /  x ]. w  e.  A  <->  w  e.  [_ z  /  x ]_ A )
1815, 17bitri 183 . . . . . 6  |-  ( [ z  /  x ]
w  e.  A  <->  w  e.  [_ z  /  x ]_ A )
1910, 14, 183bitr4g 222 . . . . 5  |-  ( [_ y  /  x ]_ A  =  [_ z  /  x ]_ A  ->  ( [ y  /  x ]
w  e.  A  <->  [ z  /  x ] w  e.  A ) )
20192alimi 1432 . . . 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 1954 . . . 4  |-  ( F/ x  w  e.  A  <->  A. y A. z ( [ y  /  x ] w  e.  A  <->  [ z  /  x ]
w  e.  A ) )
2220, 21sylibr 133 . . 3  |-  ( A. y A. z [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A  ->  F/ x  w  e.  A )
239, 22nfcd 2274 . 2  |-  ( A. y A. z [_ y  /  x ]_ A  = 
[_ z  /  x ]_ A  ->  F/_ x A )
248, 23impbii 125 1  |-  ( F/_ x A  <->  A. y A. z [_ y  /  x ]_ A  =  [_ z  /  x ]_ A )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1329    = wceq 1331   F/wnf 1436    e. wcel 1480   [wsb 1735   F/_wnfc 2266   _Vcvv 2681   [.wsbc 2904   [_csb 2998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905  df-csb 2999
This theorem is referenced by:  eusvnf  4369
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