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We report the first observation of a_{0}^{0}(980)-f_{0}(980) mixing in the decays of J/ψ→ϕf_{0}(980)→ϕa_{0}^{0}(980)→ϕηπ^{0} and χ_{c1}→a_{0}^{0}(980)π^{0}→f_{0}(980)π^{0}→π^{+}π^{-}π^{0}, using data samples of 1.31×10^{9}  J/ψ events and 4.48×10^{8}  ψ(3686) events accumulated with the BESIII detector. The signals of f_{0}(980)→a_{0}^{0}(980) and a_{0}^{0}(980)→f_{0}(980) mixing are observed at levels of statistical significance of 7.4σ and 5.5σ, respectively. The corresponding branching fractions and mixing intensities are measured and the constraint regions on the coupling constants, g_{a_{0}K^{+}K^{-}} and g_{f_{0}K^{+}K^{-}}, are estimated. The results improve the understanding of the nature of a_{0}^{0}(980) and f_{0}(980).