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Theorem eusvnf 4721
 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
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 2306 . 2
2 vex 2961 . . . . . . 7
3 nfcv 2574 . . . . . . . 8
4 nfcsb1v 3285 . . . . . . . . 9
54nfeq2 2585 . . . . . . . 8
6 csbeq1a 3261 . . . . . . . . 9
76eqeq2d 2449 . . . . . . . 8
83, 5, 7spcgf 3033 . . . . . . 7
92, 8ax-mp 5 . . . . . 6
10 vex 2961 . . . . . . 7
11 nfcv 2574 . . . . . . . 8
12 nfcsb1v 3285 . . . . . . . . 9
1312nfeq2 2585 . . . . . . . 8
14 csbeq1a 3261 . . . . . . . . 9
1514eqeq2d 2449 . . . . . . . 8
1611, 13, 15spcgf 3033 . . . . . . 7
1710, 16ax-mp 5 . . . . . 6
189, 17eqtr3d 2472 . . . . 5
1918alrimivv 1643 . . . 4
20 sbnfc2 3311 . . . 4
2119, 20sylibr 205 . . 3
2221exlimiv 1645 . 2
231, 22syl 16 1
 Colors of variables: wff set class Syntax hints:   wi 4  wal 1550  wex 1551   wceq 1653   wcel 1726  weu 2283  wnfc 2561  cvv 2958  csb 3253 This theorem is referenced by:  eusvnfb  4722  eusv2i  4723 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-sbc 3164  df-csb 3254
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