This is a very special case of the Darmois-Skitovich theorem, which deals with two
arbitrary linear forms of the independent random variables. 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.