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Theorem sbnfc2 3796
 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 3048 . . . . 5
2 csbtt 3374 . . . . 5
31, 2mpan 676 . . . 4
4 vex 3048 . . . . 5
5 csbtt 3374 . . . . 5
64, 5mpan 676 . . . 4
73, 6eqtr4d 2488 . . 3
87alrimivv 1774 . 2
9 nfv 1761 . . 3
10 eleq2 2518 . . . . . 6
11 sbsbc 3271 . . . . . . 7
12 sbcel2 3778 . . . . . . 7
1311, 12bitri 253 . . . . . 6
14 sbsbc 3271 . . . . . . 7
15 sbcel2 3778 . . . . . . 7
1614, 15bitri 253 . . . . . 6
1710, 13, 163bitr4g 292 . . . . 5
18172alimi 1685 . . . 4
19 sbnf2 2268 . . . 4
2018, 19sylibr 216 . . 3
219, 20nfcd 2587 . 2
228, 21impbii 191 1
 Colors of variables: wff setvar class Syntax hints:   wb 188  wal 1442   wceq 1444  wnf 1667  wsb 1797   wcel 1887  wnfc 2579  cvv 3045  wsbc 3267  csb 3363 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-in 3411  df-ss 3418  df-nul 3732 This theorem is referenced by:  eusvnf  4598
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