Theorem intabs 4613

Description: Absorption of a redundant conjunct in the intersection of a class abstraction. (Contributed by NM, 3-Jul-2005.)

Hypotheses

Ref	Expression
intabs.1
intabs.2
intabs.3

Assertion

Ref	Expression
intabs

Distinct variable groups:   ,   ,   ,   ,   ,

Proof of Theorem intabs

Step	Hyp	Ref	Expression
1		sseq1 3524	. . . . . 6
2		intabs.2	. . . . . 6
3	1, 2	anbi12d 710	. . . . 5
4		intabs.3	. . . . 5
5	3, 4	intmin3 4315	. . . 4
6		intnex 4609	. . . . 5
7		ssv 3523	. . . . . 6
8		sseq2 3525	. . . . . 6
9	7, 8	mpbiri 233	. . . . 5
10	6, 9	sylbi 195	. . . 4
11	5, 10	pm2.61i 164	. . 3
12		intabs.1	. . . . 5
13	12	cbvabv 2600	. . . 4
14	13	inteqi 4290	. . 3
15	11, 14	sseqtr4i 3536	. 2
16		simpr 461	. . . 4
17	16	ss2abi 3571	. . 3
18		intss 4307	. . 3
19	17, 18	ax-mp 5	. 2
20	15, 19	eqssi 3519	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  {cab 2442  cvv 3109  C_wss 3475  |^|cint 4286

This theorem is referenced by:  dfnn3  10575

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-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573

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-ne 2654  df-ral 2812  df-v 3111  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785  df-int 4287