If the consequence is not true, then the implication fails, regardless of the truth of the antecedent.
Discussion
Implication?
If by implication you mean the consequence, then yes. That is a tautological equivalence.
If by implication you mean the theorem, the "if A --> C," then no that doesn't follow. If C is false, then A is false or "A=>C" is false.
I forgot, this is a biconditional but what I said above still applies. It just applies both directions.