Suppose that you have shown that whenever $X$ is true, then $Y$ is true, and whenever $X$ is false, then $Y$ is false. Have you now demonstrated that $X$ and $Y$ are logically equivalent? Explain.
To show that $X$ and $Y$ are logically equivalent, $X\iff Y$, we need to show both $X\implies Y$ and $Y\implies X$.
Demonstrating that whenever $X$ is true, then $Y$ is true, is equivalent to demonstrating $X\implies Y$.
We're given that whenever $X$ is false, then $Y$ is false. This $\neg X\implies\neg Y$ is logically equivalent to $Y\implies X$.
So we have demonstrated that $X$ and $Y$ are indeed logically equivalent.
