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Theorem eusvnf 4450
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 2056 . 2  |-  ( E! y A. x  y  =  A  ->  E. y A. x  y  =  A )
2 vex 2740 . . . . . . 7  |-  z  e. 
_V
3 nfcv 2319 . . . . . . . 8  |-  F/_ x
z
4 nfcsb1v 3090 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ A
54nfeq2 2331 . . . . . . . 8  |-  F/ x  y  =  [_ z  /  x ]_ A
6 csbeq1a 3066 . . . . . . . . 9  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
76eqeq2d 2189 . . . . . . . 8  |-  ( x  =  z  ->  (
y  =  A  <->  y  =  [_ z  /  x ]_ A ) )
83, 5, 7spcgf 2819 . . . . . . 7  |-  ( z  e.  _V  ->  ( A. x  y  =  A  ->  y  =  [_ z  /  x ]_ A
) )
92, 8ax-mp 5 . . . . . 6  |-  ( A. x  y  =  A  ->  y  =  [_ z  /  x ]_ A )
10 vex 2740 . . . . . . 7  |-  w  e. 
_V
11 nfcv 2319 . . . . . . . 8  |-  F/_ x w
12 nfcsb1v 3090 . . . . . . . . 9  |-  F/_ x [_ w  /  x ]_ A
1312nfeq2 2331 . . . . . . . 8  |-  F/ x  y  =  [_ w  /  x ]_ A
14 csbeq1a 3066 . . . . . . . . 9  |-  ( x  =  w  ->  A  =  [_ w  /  x ]_ A )
1514eqeq2d 2189 . . . . . . . 8  |-  ( x  =  w  ->  (
y  =  A  <->  y  =  [_ w  /  x ]_ A ) )
1611, 13, 15spcgf 2819 . . . . . . 7  |-  ( w  e.  _V  ->  ( A. x  y  =  A  ->  y  =  [_ w  /  x ]_ A
) )
1710, 16ax-mp 5 . . . . . 6  |-  ( A. x  y  =  A  ->  y  =  [_ w  /  x ]_ A )
189, 17eqtr3d 2212 . . . . 5  |-  ( A. x  y  =  A  ->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
1918alrimivv 1875 . . . 4  |-  ( A. x  y  =  A  ->  A. z A. w [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
20 sbnfc2 3117 . . . 4  |-  ( F/_ x A  <->  A. z A. w [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
2119, 20sylibr 134 . . 3  |-  ( A. x  y  =  A  -> 
F/_ x A )
2221exlimiv 1598 . 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 1351    = wceq 1353   E.wex 1492   E!weu 2026    e. wcel 2148   F/_wnfc 2306   _Vcvv 2737   [_csb 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963  df-csb 3058
This theorem is referenced by:  eusvnfb  4451  eusv2i  4452
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