Thompson stepped down as Yahoo chief last week, 10 days after activist investor Daniel Loeb accused him of padding his biography by faking a computer science degree.
Yahoo did confirm Loeb's allegations, calling the discrepancy in the education records an "inadvertent error."
F5 did not cite a reason for Thompson's exit. He had been an independent director on F5's board for 4 years.
(Reporting by Himank Sharma in Bangalore; Editing by Viraj Nair)
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