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The notion of (regular) left (θ,ϕ)-derivations of a BCI-algebra is introduced, some useful examples are discussed, and related properties are investigated. Conditions for a left (θ,ϕ)-derivation to be regular are provided. The concepts of a d(θ,ϕ)-invariant left (θ,ϕ)-derivation and θ-ideal are introduced, and their relations are discussed. Furthermore, some more interesting results are established.