Returning cell content from array based on two conditions

Hi folks. New to site, glad to have the resource and everyone's help. I have a spreadsheet with patient ID numbers, admission dates and diagnoses, among other information. I've been struggling to come up with the right functions and nesting to return text from a cell based on two conditions being met:
1. Locate in an array an ID# from a reference cell.
2. Then locate the admission date from another reference cell.
3. Go to the row specified from 1. and 2. and the column for diagnosis.
4. Return the text for that diagnosis for the patient ID# and admission specified.
I've tried VLOOKUP, INDEX, ROW and others in various nesting combinations and am getting nowhere. I've sorted by ID#, then ADMISSION date. I've attached an example of the spreadsheet. It contains many patients with many different admissions, some may have only one admission, others 7 or more. It basically looks as follows:
ID# ADMISSION DIAGNOSIS
1234 1/5/09 CHF
1234 2/30/09 AFIB
1234 5/1/10 CHEST PAIN
3333 5/5/09 DM
4321 1/4/09 CKD
4321 10/710 AMS
etc.

This is great! Love this place...thanks.

You both just did in a minute what I couldn't do in hours. Thanks so much. Now I will try to understand the formula.....