Theorem csbifgOLD 3992

Description: Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by Mario Carneiro, 14-Nov-2016.) Obsolete as of 19-Aug-2018. Use csbif 3991 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
csbifgOLD

Proof of Theorem csbifgOLD

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3437	. . 3
2		dfsbcq2 3330	. . . 4
3		csbeq1 3437	. . . 4
4		csbeq1 3437	. . . 4
5	2, 3, 4	ifbieq12d 3968	. . 3
6	1, 5	eqeq12d 2479	. 2
7		vex 3112	. . 3
8		nfs1v 2181	. . . 4
9		nfcsb1v 3450	. . . 4
10		nfcsb1v 3450	. . . 4
11	8, 9, 10	nfif 3970	. . 3
12		sbequ12 1992	. . . 4
13		csbeq1a 3443	. . . 4
14		csbeq1a 3443	. . . 4
15	12, 13, 14	ifbieq12d 3968	. . 3
16	7, 11, 15	csbief 3459	. 2
17	6, 16	vtoclg 3167	1

Syntax hints:  ->wi 4  =wceq 1395  [wsb 1739  e.wcel 1818  [.wsbc 3327  [_csb 3434  ifcif 3941

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-un 3480  df-if 3942