Theorem sbcel2gOLD 3832

Description: Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) Obsolete as of 18-Aug-2018. Use sbcel2 3831 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbcel2gOLD

Distinct variable group:   ,

Proof of Theorem sbcel2gOLD

Step	Hyp	Ref	Expression
1		sbcel12gOLD 3824	. 2
2		csbconstg 3447	. . 3
3	2	eleq1d 2526	. 2
4	1, 3	bitrd 253	1

Syntax hints:  ->wi 4  <->wb 184  e.wcel 1818  [.wsbc 3327  [_csb 3434

This theorem is referenced by:  csbcomgOLD  3838  sbccsbgOLD  3850  csbabgOLD  3856  csbunigOLD  4278  csbxpgOLD  5087  csbrngOLD  5474  sbcssOLD  33313  sbcssgVD  33683  csbingVD  33684  csbunigVD  33698

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