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Theorem sbnfc2 3683
 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 2965 . . . . 5
2 csbtt 3277 . . . . 5
31, 2mpan 653 . . . 4
4 vex 2965 . . . . 5
5 csbtt 3277 . . . . 5
64, 5mpan 653 . . . 4
73, 6eqtr4d 2477 . . 3
87alrimivv 1643 . 2
9 nfv 1630 . . 3
10 eleq2 2503 . . . . . 6
11 sbsbc 3171 . . . . . . 7
12 sbcel2gOLD 3659 . . . . . . . 8
131, 12ax-mp 5 . . . . . . 7
1411, 13bitri 242 . . . . . 6
15 sbsbc 3171 . . . . . . 7
16 sbcel2gOLD 3659 . . . . . . . 8
174, 16ax-mp 5 . . . . . . 7
1815, 17bitri 242 . . . . . 6
1910, 14, 183bitr4g 281 . . . . 5
20192alimi 1570 . . . 4
21 sbnf2 2190 . . . 4
2220, 21sylibr 205 . . 3
239, 22nfcd 2573 . 2
248, 23impbii 182 1
 Colors of variables: wff set class Syntax hints:   wb 178  wal 1550  wnf 1554   wceq 1653  wsb 1659   wcel 1727  wnfc 2565  cvv 2962  wsbc 3167  csb 3267 This theorem is referenced by:  eusvnf  4747 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 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423 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-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-v 2964  df-sbc 3168  df-csb 3268
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