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Theorem sbnfc2 3854
Description: Two ways of expressing " is (effectively) not free in ." (Contributed by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
sbnfc2
Distinct variable groups:   , ,   , ,

Proof of Theorem sbnfc2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 3112 . . . . 5
2 csbtt 3445 . . . . 5
31, 2mpan 670 . . . 4
4 vex 3112 . . . . 5
5 csbtt 3445 . . . . 5
64, 5mpan 670 . . . 4
73, 6eqtr4d 2501 . . 3
87alrimivv 1720 . 2
9 nfv 1707 . . 3
10 eleq2 2530 . . . . . 6
11 sbsbc 3331 . . . . . . 7
12 sbcel2 3831 . . . . . . 7
1311, 12bitri 249 . . . . . 6
14 sbsbc 3331 . . . . . . 7
15 sbcel2 3831 . . . . . . 7
1614, 15bitri 249 . . . . . 6
1710, 13, 163bitr4g 288 . . . . 5
18172alimi 1634 . . . 4
19 sbnf2 2183 . . . 4
2018, 19sylibr 212 . . 3
219, 20nfcd 2613 . 2
228, 21impbii 188 1
Colors of variables: wff setvar class
Syntax hints:  <->wb 184  A.wal 1393  =wceq 1395  F/wnf 1616  [wsb 1739  e.wcel 1818  F/_wnfc 2605   cvv 3109  [.wsbc 3327  [_csb 3434
This theorem is referenced by:  eusvnf  4647
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-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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