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.

Step	Hyp	Ref	Expression
1		vex 3112	. . . . 5
2		csbtt 3445	. . . . 5
3	1, 2	mpan 670	. . . 4
4		vex 3112	. . . . 5
5		csbtt 3445	. . . . 5
6	4, 5	mpan 670	. . . 4
7	3, 6	eqtr4d 2501	. . 3
8	7	alrimivv 1720	. 2
9		nfv 1707	. . 3
10		eleq2 2530	. . . . . 6
11		sbsbc 3331	. . . . . . 7
12		sbcel2 3831	. . . . . . 7
13	11, 12	bitri 249	. . . . . 6
14		sbsbc 3331	. . . . . . 7
15		sbcel2 3831	. . . . . . 7
16	14, 15	bitri 249	. . . . . 6
17	10, 13, 16	3bitr4g 288	. . . . 5
18	17	2alimi 1634	. . . 4
19		sbnf2 2183	. . . 4
20	18, 19	sylibr 212	. . 3
21	9, 20	nfcd 2613	. 2
22	8, 21	impbii 188	1

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