This article addresses a general criterion providing a zero duality gap for convex programs in the setting of the real locally convex spaces. The main theorem of our work is formulated only in terms of the constraints of the program, hence it holds true for any objective function fulfilling a very general qualification condition, implied for instance by standard qualification criteria of Moreau-Rockafellar or Attouch-Br ́ezis type. This result generalizes recent theorems by Champion, Ban & Song and Jeyakumar & Li.