Theorem iinin2 4400

Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4384 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)

Assertion

Ref	Expression
iinin2

Distinct variable groups:   ,   ,

Proof of Theorem iinin2

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.28zv 3924	. . . 4
2		elin 3686	. . . . 5
3	2	ralbii 2888	. . . 4
4		vex 3112	. . . . . 6
5		eliin 4336	. . . . . 6
6	4, 5	ax-mp 5	. . . . 5
7	6	anbi2i 694	. . . 4
8	1, 3, 7	3bitr4g 288	. . 3
9		eliin 4336	. . . 4
10	4, 9	ax-mp 5	. . 3
11		elin 3686	. . 3
12	8, 10, 11	3bitr4g 288	. 2
13	12	eqrdv 2454	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807  cvv 3109  i^icin 3474  c0 3784  |^|_ciin 4331

This theorem is referenced by:  iinin1  4401

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-ne 2654  df-ral 2812  df-v 3111  df-dif 3478  df-in 3482  df-nul 3785  df-iin 4333