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The aim of this paper is to prove the two-variable anticyclotomic Iwasawamain conjecture for Hida families and a definite version of the horizontalnon-vanishing conjecture, which are formulated in Longo-Vigni. Our approach isbased on the two-variable anticyclotomic control theorem for Selmer groups forHida families and the relation between the two-variable anticyclotomic$L$-function for Hida families built out of $p$-adic families of Gross pointson definite Shimura curves studied in Castella-Longo and Castella-Kim-Longo andthe self-dual twist of the specialisation to the anticyclotomic line of thethree-variable $p$-adic $L$-function of Skinner-Urban.

Anticyclotomic main conjecture and the non-triviality of Rankin-Selberg $L$-values in Hida families