More on Implications

Recall: $P \implies Q$ is a statement built from statements.

Now: $P(x) \implies Q(x)$ is a statement built from open sentences. This is the form of most theorems.

Example:

$$\begin{align} P(x)&: x = -3\\ Q(x)&: |x| = 3 \end{align}\\ \text{If } x = -3, \text{ then } |x| = 3.$$