Theorem intab 4317

Description: The intersection of a special case of a class abstraction. may be free in and , which can be thought of a () and A(). Typically, abrexex2 6781 or abexssex 6782 can be used to satisfy the second hypothesis. (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

Hypotheses

Ref	Expression
intab.1
intab.2

Assertion

Ref	Expression
intab

Distinct variable groups:   ,   ,   ,

Proof of Theorem intab

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2461	. . . . . . . . . 10
2	1	anbi2d 703	. . . . . . . . 9
3	2	exbidv 1714	. . . . . . . 8
4	3	cbvabv 2600	. . . . . . 7
5		intab.2	. . . . . . 7
6	4, 5	eqeltri 2541	. . . . . 6
7		nfe1 1840	. . . . . . . . 9
8	7	nfab 2623	. . . . . . . 8
9	8	nfeq2 2636	. . . . . . 7
10		eleq2 2530	. . . . . . . 8
11	10	imbi2d 316	. . . . . . 7
12	9, 11	albid 1885	. . . . . 6
13	6, 12	elab 3246	. . . . 5
14		19.8a 1857	. . . . . . . . 9
15	14	ex 434	. . . . . . . 8
16	15	alrimiv 1719	. . . . . . 7
17		intab.1	. . . . . . . 8
18	17	sbc6 3354	. . . . . . 7
19	16, 18	sylibr 212	. . . . . 6
20		df-sbc 3328	. . . . . 6
21	19, 20	sylib 196	. . . . 5
22	13, 21	mpgbir 1622	. . . 4
23		intss1 4301	. . . 4
24	22, 23	ax-mp 5	. . 3
25		19.29r 1684	. . . . . . . 8
26		simplr 755	. . . . . . . . . 10
27		pm3.35 587	. . . . . . . . . . 11
28	27	adantlr 714	. . . . . . . . . 10
29	26, 28	eqeltrd 2545	. . . . . . . . 9
30	29	exlimiv 1722	. . . . . . . 8
31	25, 30	syl 16	. . . . . . 7
32	31	ex 434	. . . . . 6
33	32	alrimiv 1719	. . . . 5
34		vex 3112	. . . . . 6
35	34	elintab 4297	. . . . 5
36	33, 35	sylibr 212	. . . 4
37	36	abssi 3574	. . 3
38	24, 37	eqssi 3519	. 2
39	38, 4	eqtri 2486	1

Syntax hints:  ->wi 4  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  cvv 3109  [.wsbc 3327  C_wss 3475  |^|cint 4286

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-in 3482  df-ss 3489  df-int 4287