Theorem iineq2 4348

Description: Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iineq2

Proof of Theorem iineq2

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2530	. . . . 5
2	1	ralimi 2850	. . . 4
3		ralbi 2988	. . . 4
4	2, 3	syl 16	. . 3
5	4	abbidv 2593	. 2
6		df-iin 4333	. 2
7		df-iin 4333	. 2
8	5, 6, 7	3eqtr4g 2523	1

Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  e.wcel 1818  {cab 2442  A.wral 2807  |^|_ciin 4331

This theorem is referenced by:  iineq2i  4350  iineq2d  4351  firest  14830  iincld  19540  elrfirn2  30628

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-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-ral 2812  df-iin 4333