Theorem csbxpgOLD 5087

Description: Distribute proper substitution through the Cartesian product of two classes. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbrn 5473 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
csbxpgOLD

Proof of Theorem csbxpgOLD

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

Step	Hyp	Ref	Expression
1		csbabgOLD 3856	. . 3
2		sbcexgOLD 3382	. . . . 5
3		sbcexgOLD 3382	. . . . . . 7
4		sbcangOLD 3371	. . . . . . . . 9
5		sbcg 3401	. . . . . . . . . 10
6		sbcangOLD 3371	. . . . . . . . . . 11
7		sbcel2gOLD 3832	. . . . . . . . . . . 12
8		sbcel2gOLD 3832	. . . . . . . . . . . 12
9	7, 8	anbi12d 710	. . . . . . . . . . 11
10	6, 9	bitrd 253	. . . . . . . . . 10
11	5, 10	anbi12d 710	. . . . . . . . 9
12	4, 11	bitrd 253	. . . . . . . 8
13	12	exbidv 1714	. . . . . . 7
14	3, 13	bitrd 253	. . . . . 6
15	14	exbidv 1714	. . . . 5
16	2, 15	bitrd 253	. . . 4
17	16	abbidv 2593	. . 3
18	1, 17	eqtrd 2498	. 2
19		df-xp 5010	. . . 4
20		df-opab 4511	. . . 4
21	19, 20	eqtri 2486	. . 3
22	21	csbeq2i 3836	. 2
23		df-xp 5010	. . 3
24		df-opab 4511	. . 3
25	23, 24	eqtri 2486	. 2
26	18, 22, 25	3eqtr4g 2523	1

Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  [.wsbc 3327  [_csb 3434  <.cop 4035  {copab 4509  X.cxp 5002

This theorem is referenced by:  csbresgOLD  5282  csbresgVD  33695

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  df-opab 4511  df-xp 5010