||This paper presents the notion of D-normal proofs,
which is defined syntactically and gives
one of the weakest condition for uniqueness of normal proofs.
This paper proves the following results:
(1) beta eta D-normal proofs of a formula are unique.
(2) A beta-normal proof of a PNN-formula is D-normal.
(3) A beta-normal proof of a minimal formula in BCK logic is D-normal.
These results give other proofs of
uniqueness of beta eta-normal proofs of a PNN-formula,
and uniqueness of beta eta-normal proofs of a minimal formula in BCK logic.