# Is the Formula Logically Valid?

I have a question for my exam and I find it hard to understand.

I have to prove that the following formula is logically valid:

The professor told me to "push" all the symbols inside the brackets, and use the deduction theorem.

But I don't know how to do it, because I can't find the identities to push the "exist" symbol inside the brackets.

Your help is appriciated, thank you.

Alan

• See the drinker's paradox. – Git Gud Feb 11 '15 at 21:20
• That's very interesting! Thanks, I'll have a look. – Alan Feb 11 '15 at 21:28
• See also this, this comment and this thread. – Git Gud Feb 11 '15 at 21:30

Assume $\forall yp(y)$. Then $p(x)\to\forall yp(y)$ is true for any $x$. If on th eother hand $\neg \forall yp(y)$, then $\exists y\neg p(y)$. Let $x$ be such an $y$ then again $p(x)\to\forall yp(y)$ is true, this time because the antecedent is false.
I hope this helps $\ddot\smile$