Theorem csbcomgOLD 3838

Description: Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.) Obsolete as of 18-Aug-2018. Use csbcom 3837 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
csbcomgOLD

Distinct variable groups:   ,   ,   ,

Proof of Theorem csbcomgOLD

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3118	. 2
2		elex 3118	. 2
3		sbccom 3407	. . . . . 6
4	3	a1i 11	. . . . 5
5		sbcel2gOLD 3832	. . . . . . 7
6	5	sbcbidv 3386	. . . . . 6
7	6	adantl 466	. . . . 5
8		sbcel2gOLD 3832	. . . . . . 7
9	8	sbcbidv 3386	. . . . . 6
10	9	adantr 465	. . . . 5
11	4, 7, 10	3bitr3d 283	. . . 4
12		sbcel2gOLD 3832	. . . . 5
13	12	adantr 465	. . . 4
14		sbcel2gOLD 3832	. . . . 5
15	14	adantl 466	. . . 4
16	11, 13, 15	3bitr3d 283	. . 3
17	16	eqrdv 2454	. 2
18	1, 2, 17	syl2an 477	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  cvv 3109  [.wsbc 3327  [_csb 3434

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