Note the potential for an out-of-range index with "i+3". You could do something like:
with open("file.txt", "r") as f:
searchlines = f.readlines()
j=len(searchlines)-1
for i, line in enumerate(searchlines):
if "searchphrase" in line:
k=min(i+3,j)
for l in searchlines[i:k]: print l,
print
Edit: maybe not necessary. I just tested some examples. x[y] will give errors if y is out of range, but x[y:z] doesn't seem to give errors for out of range values of y and z.