Theorem sbcel12gOLD 3824

Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcel12 3823 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbcel12gOLD

Proof of Theorem sbcel12gOLD

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3330	. . 3
2		dfsbcq2 3330	. . . . 5
3	2	abbidv 2593	. . . 4
4		dfsbcq2 3330	. . . . 5
5	4	abbidv 2593	. . . 4
6	3, 5	eleq12d 2539	. . 3
7		nfs1v 2181	. . . . . 6
8	7	nfab 2623	. . . . 5
9		nfs1v 2181	. . . . . 6
10	9	nfab 2623	. . . . 5
11	8, 10	nfel 2632	. . . 4
12		sbab 2604	. . . . 5
13		sbab 2604	. . . . 5
14	12, 13	eleq12d 2539	. . . 4
15	11, 14	sbie 2149	. . 3
16	1, 6, 15	vtoclbg 3168	. 2
17		df-csb 3435	. . 3
18		df-csb 3435	. . 3
19	17, 18	eleq12i 2536	. 2
20	16, 19	syl6bbr 263	1

Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  [wsb 1739  e.wcel 1818  {cab 2442  [.wsbc 3327  [_csb 3434

This theorem is referenced by:  sbcel2gOLD  3832  sbccsb2gOLD  3852  csbxpgVD  33694  csbrngVD  33696

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-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