Logical Equivalences with quantifiers

Hello

I am running into problems with the following:

∃**x(P (x) ⇒ (Q(x) ⇒ R(x)))⇐⇒ (¬∀xP (x) ∨ ¬∀yQ(y) ∨ ∃zR(z))**

Although I do not master logical equivalences, I have been able to solve some earlier. However, since this problem has quantifiers, I don't know how to approach it.

Any help is highly appreciated.

Re: Logical Equivalences with quantifiers

P(x) => (Q(x) => R(x)) is equivalent to ~P(x) \/ ~Q(x) \/ R(x).

Existential quantifier distributes over disjunction (because existential quantifier is basically a disjunction over all elements of the domain): $\displaystyle \exists x\,(A(x)\lor B(x))\iff(\exists x\,A(x))\lor(\exists x\,B(x))$.

Next, existential quantifier changes into universal quantifier and vice versa when it is moved through a negation: $\displaystyle \exists x\,(\neg A(x))\iff\neg(\forall x\,A(x))$.

These equivalences are sufficient to derive the one you need.