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## Homework Statement

What is the first step in proving a proposition of the form ##\forall x (P(x) \implies Q(x))##

## Homework Equations

## The Attempt at a Solution

So this isn't exactly a homework question, but I am just trying to figure things out. So say that we have a conjecture of the form ##\forall x (P(x) \implies Q(x))##. In my textbook, it says that to (formally) prove a proposition such as this, we first prove ##P(c) \implies Q(c)##, where c is an arbitrary element of the domain of discourse, and then by the inference rule of universal generalization, conclude that ##\forall x (P(x) \implies Q(x))##.

However, I confused as to how to get to the second step. First we begin with ##\forall x (P(x) \implies Q(x))##. So to get to ##P(c) \implies Q(c)## don't we need to apply universal instantiation? And to apply universal instantiation, don't we first need to know that ##\forall x (P(x) \implies Q(x))## is true? Isn't that kind of circular?