Prove that ∀y(∀xP(x, y) → ∃xP(x, y)) is a tautology

Question

So I'm attempting to prove the above proposition is a tautology, but am struggling to structure my argument as a logical proof.

So my current thinking is that there exist a few situations, regardless of the values of x,y

• P(x,y) is always true, in which case true => true, which is true
• P(x,y) is always false, in which case false => false, which is true
• P(x,y) is a mix of true and false, in which case there must always be an x, such that P(x,y) is true, in which case we have false => true, which is true

So I can see that the proposition is a tautology, however, I am having trouble wording this a logical proof. How would you recommend I do this?

Answer 1

Your proof needs only show that:

Taking any arbitrary $y$, should every $x$ satisfies $P(x,y)$, then (because $\underline{\hspace{10ex}}$) there will be some $x$ which satisfies $P(x,y)$ .

$$\therefore\quad\forall y~(\forall x~P(x,y)\to\exists x~P(x,y))$$

Just fill in the blank.