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Theorem eusvnf 4647
 Description: Even if is free in , 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
Distinct variable groups:   ,   ,

Proof of Theorem eusvnf
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 euex 2308 . 2
2 vex 3112 . . . . . . 7
3 nfcv 2619 . . . . . . . 8
4 nfcsb1v 3450 . . . . . . . . 9
54nfeq2 2636 . . . . . . . 8
6 csbeq1a 3443 . . . . . . . . 9
76eqeq2d 2471 . . . . . . . 8
83, 5, 7spcgf 3189 . . . . . . 7
92, 8ax-mp 5 . . . . . 6
10 vex 3112 . . . . . . 7
11 nfcv 2619 . . . . . . . 8
12 nfcsb1v 3450 . . . . . . . . 9
1312nfeq2 2636 . . . . . . . 8
14 csbeq1a 3443 . . . . . . . . 9
1514eqeq2d 2471 . . . . . . . 8
1611, 13, 15spcgf 3189 . . . . . . 7
1710, 16ax-mp 5 . . . . . 6
189, 17eqtr3d 2500 . . . . 5
1918alrimivv 1720 . . . 4
20 sbnfc2 3854 . . . 4
2119, 20sylibr 212 . . 3
2221exlimiv 1722 . 2
231, 22syl 16 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  E!weu 2282  F/_wnfc 2605   cvv 3109  [_csb 3434 This theorem is referenced by:  eusvnfb  4648  eusv2i  4649 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-fal 1401  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785
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