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Theorem eusvnf 4185
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 1930 . 2  |-  ( E! y A. x  y  =  A  ->  E. y A. x  y  =  A )
2 vex 2560 . . . . . . 7  |-  z  e. 
_V
3 nfcv 2178 . . . . . . . 8  |-  F/_ x
z
4 nfcsb1v 2882 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ A
54nfeq2 2189 . . . . . . . 8  |-  F/ x  y  =  [_ z  /  x ]_ A
6 csbeq1a 2860 . . . . . . . . 9  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
76eqeq2d 2051 . . . . . . . 8  |-  ( x  =  z  ->  (
y  =  A  <->  y  =  [_ z  /  x ]_ A ) )
83, 5, 7spcgf 2635 . . . . . . 7  |-  ( z  e.  _V  ->  ( A. x  y  =  A  ->  y  =  [_ z  /  x ]_ A
) )
92, 8ax-mp 7 . . . . . 6  |-  ( A. x  y  =  A  ->  y  =  [_ z  /  x ]_ A )
10 vex 2560 . . . . . . 7  |-  w  e. 
_V
11 nfcv 2178 . . . . . . . 8  |-  F/_ x w
12 nfcsb1v 2882 . . . . . . . . 9  |-  F/_ x [_ w  /  x ]_ A
1312nfeq2 2189 . . . . . . . 8  |-  F/ x  y  =  [_ w  /  x ]_ A
14 csbeq1a 2860 . . . . . . . . 9  |-  ( x  =  w  ->  A  =  [_ w  /  x ]_ A )
1514eqeq2d 2051 . . . . . . . 8  |-  ( x  =  w  ->  (
y  =  A  <->  y  =  [_ w  /  x ]_ A ) )
1611, 13, 15spcgf 2635 . . . . . . 7  |-  ( w  e.  _V  ->  ( A. x  y  =  A  ->  y  =  [_ w  /  x ]_ A
) )
1710, 16ax-mp 7 . . . . . 6  |-  ( A. x  y  =  A  ->  y  =  [_ w  /  x ]_ A )
189, 17eqtr3d 2074 . . . . 5  |-  ( A. x  y  =  A  ->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
1918alrimivv 1755 . . . 4  |-  ( A. x  y  =  A  ->  A. z A. w [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
20 sbnfc2 2906 . . . 4  |-  ( F/_ x A  <->  A. z A. w [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
2119, 20sylibr 137 . . 3  |-  ( A. x  y  =  A  -> 
F/_ x A )
2221exlimiv 1489 . 2  |-  ( E. y A. x  y  =  A  ->  F/_ x A )
231, 22syl 14 1  |-  ( E! y A. x  y  =  A  ->  F/_ x A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1241    = wceq 1243   E.wex 1381    e. wcel 1393   E!weu 1900   F/_wnfc 2165   _Vcvv 2557   [_csb 2852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853
This theorem is referenced by:  eusvnfb  4186  eusv2i  4187
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