Theorem sbcbrgOLD 4504

Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) Obsolete as of 22-Aug-2018. Use sbcbr123 4503 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbcbrgOLD

Proof of Theorem sbcbrgOLD

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3330	. 2
2		csbeq1 3437	. . 3
3		csbeq1 3437	. . 3
4		csbeq1 3437	. . 3
5	2, 3, 4	breq123d 4466	. 2
6		nfcsb1v 3450	. . . 4
7		nfcsb1v 3450	. . . 4
8		nfcsb1v 3450	. . . 4
9	6, 7, 8	nfbr 4496	. . 3
10		csbeq1a 3443	. . . 4
11		csbeq1a 3443	. . . 4
12		csbeq1a 3443	. . . 4
13	10, 11, 12	breq123d 4466	. . 3
14	9, 13	sbie 2149	. 2
15	1, 5, 14	vtoclbg 3168	1

Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  [wsb 1739  e.wcel 1818  [.wsbc 3327  [_csb 3434   class class class wbr 4452

This theorem is referenced by:  csbfv12gOLD  5907

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-3an 975  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-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453