The straightforward elimination of union types is known to break subject reduction, and for some extensions of the lambda-calculus, to break strong normalization as well. Similarly, the straightforward elim- ination of implicit existential types breaks subject reduction. We propose elimination rules for union types and implicit existential quantification which use a form call-by-value issued from Girard's re- ducibility candidates. We show that these rules remedy the above men- tioned difficulties, for strong normalization and, for the existential quan- tification, for subject reduction as well. Moreover, for extensions of the lambda-calculus based on intuitionistic logic, we show that the obtained existential quantification is equivalent to its usual impredicative encoding w.r.t. provability in realizability models built from reducibility candidates and biorthogonals.