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Theorem sbnfc2 2906
 Description: Two ways of expressing " is (effectively) not free in ." (Contributed by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
sbnfc2
Distinct variable groups:   ,,   ,,
Allowed substitution hint:   ()

Proof of Theorem sbnfc2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 2560 . . . . 5
2 csbtt 2862 . . . . 5
31, 2mpan 400 . . . 4
4 vex 2560 . . . . 5
5 csbtt 2862 . . . . 5
64, 5mpan 400 . . . 4
73, 6eqtr4d 2075 . . 3
87alrimivv 1755 . 2
9 nfv 1421 . . 3
10 eleq2 2101 . . . . . 6
11 sbsbc 2768 . . . . . . 7
12 sbcel2g 2871 . . . . . . . 8
131, 12ax-mp 7 . . . . . . 7
1411, 13bitri 173 . . . . . 6
15 sbsbc 2768 . . . . . . 7
16 sbcel2g 2871 . . . . . . . 8
174, 16ax-mp 7 . . . . . . 7
1815, 17bitri 173 . . . . . 6
1910, 14, 183bitr4g 212 . . . . 5
20192alimi 1345 . . . 4
21 sbnf2 1857 . . . 4
2220, 21sylibr 137 . . 3
239, 22nfcd 2173 . 2
248, 23impbii 117 1
 Colors of variables: wff set class Syntax hints:   wb 98  wal 1241   wceq 1243  wnf 1349   wcel 1393  wsb 1645  wnfc 2165  cvv 2557  wsbc 2764  csb 2852 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853 This theorem is referenced by:  eusvnf  4185
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