Simple math logic question: $A$ is $B$ if $C$, $A$ is $B$ if $D$ then $C \Leftrightarrow D$? Suppose that $A$ is $B$ if $C$, $A$ is $B$ if $D$ then $C \Leftrightarrow D$?
Full context: I am looking over something which stated
Definition: a matrix $A$ is "Hurwitz-Cantor" if condition $C$ is satisfied.
Then it follows by Theorem:
A matrix $A$ is "Hurwitz-Cantor" if $D$ is satisfied.
Then somewhere in a proof much later on, the authors seem to assert that: Since $D$ is satisfied, therefore $A$ is Hurwitz-Cantor, and thus $C$ is satisfied.
To me, the logic is off. Can anyone check?

In a definition of a qualifying adjective, "if" usually really means "if and only if".
But it looks like your condition $D$ is only sufficient, not necessary. So I would say that $D\implies C$, but not necessarily that $C\implies D$.
