subsets

require(data.table)
DT = as.data.table(df)

# 1. Patients with ah and ihd
DT[,.SD["ah"%in%diagnosis && "ihd"%in%diagnosis],by=id]

     id diagnosis
[1,]  2        ah
[2,]  2       ihd
[3,]  2        im
[4,]  4        ah
[5,]  4       ihd
[6,]  4    angina

# 2. Patients with ah but no ihd
DT[,.SD["ah"%in%diagnosis && !"ihd"%in%diagnosis],by=id]

     id diagnosis
[1,]  1        ah
[2,]  3        ah
[3,]  3    stroke


# 3. Patients with  ihd but no ah?
DT[,.SD[!"ah"%in%diagnosis && "ihd"%in%diagnosis],by=id]

     id diagnosis
[1,]  5       ihd

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