This is a very special case of the Darmois-Skitovich theorem, which deals with two arbitrary linear forms of the independent random variables, without any a priori conditions about their distributions. See, for example
Kagan, A.M., Linnik, Yu.V.; Rao, C.Radhakrishna Characterization problems in mathematical statistics,
or perhaps more accessible
Theory Probab. Appl., 57(3), 368–374. I. A. Ibragimov, On the Skitovich--Darmois--Ramachandran Theorem Theory Probab. Appl., 57(3), 368–374.
Or just type "Darmois-Skitovich theorem" on Google.