Disjunctive elimination rule and its application in MTL

Firstly, the disjunctive elimination rule is proved valid in Esteva and Godo's MTL and Hájek's BL logics. Secondly, from this rule a general method is presented for finitely axiomatizing the intersection of the sets of theorems of two given schematic extensions of MTL or BL. Thirdly, the axiomatic system of NM is simplified by replacing the axiom schemata (NM) with another which is built up with one variable. Fourthly, the intersection of the sets of tautologies of NM and G, NM and Π, NM and Ł are finitely axiomatized by our method. Finally, new axiom systems for ŁG, ŁΠ, ΠG, ŁΠG are presented by our method and some comments are made on the relations between these systems and those given by Cignoli et al.