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Theorem eusvnf 4151
Description: Even if is free in , it is effectively bound when is single-valued. (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
eusvnf  F/_
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem eusvnf
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 euex 1927 . 2
2 vex 2554 . . . . . . 7 
_V
3 nfcv 2175 . . . . . . . 8  F/_
4 nfcsb1v 2876 . . . . . . . . 9  F/_ [_  ]_
54nfeq2 2186 . . . . . . . 8  F/  [_  ]_
6 csbeq1a 2854 . . . . . . . . 9  [_  ]_
76eqeq2d 2048 . . . . . . . 8 
[_  ]_
83, 5, 7spcgf 2629 . . . . . . 7  _V  [_  ]_
92, 8ax-mp 7 . . . . . 6  [_  ]_
10 vex 2554 . . . . . . 7 
_V
11 nfcv 2175 . . . . . . . 8  F/_
12 nfcsb1v 2876 . . . . . . . . 9  F/_ [_  ]_
1312nfeq2 2186 . . . . . . . 8  F/  [_  ]_
14 csbeq1a 2854 . . . . . . . . 9  [_  ]_
1514eqeq2d 2048 . . . . . . . 8 
[_  ]_
1611, 13, 15spcgf 2629 . . . . . . 7  _V  [_  ]_
1710, 16ax-mp 7 . . . . . 6  [_  ]_
189, 17eqtr3d 2071 . . . . 5  [_  ]_  [_  ]_
1918alrimivv 1752 . . . 4  [_  ]_  [_  ]_
20 sbnfc2 2900 . . . 4  F/_  [_  ]_  [_  ]_
2119, 20sylibr 137 . . 3  F/_
2221exlimiv 1486 . 2  F/_
231, 22syl 14 1  F/_
Colors of variables: wff set class
Syntax hints:   wi 4  wal 1240   wceq 1242  wex 1378   wcel 1390  weu 1897   F/_wnfc 2162   _Vcvv 2551   [_csb 2846
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847
This theorem is referenced by:  eusvnfb  4152  eusv2i  4153
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