Imply implies equivalence

Question

Let $F_1$ : $\forall x \forall y (P(x) \lor \neg P(y))$, and $F_2$ : $\forall x \forall y (P(x) \leftrightarrow P(y))$. Does $F_1 \rightarrow F_2$? I know it wouldn't if we did not have any quantifiers but whats's the case when we have quatifiers as well? If it's false then give a model where it is false.

Answer 1

• $P(x)\lor \neg P(y)$ can be rewritten as $P(y)\to P(x)$.
• Note that $\forall x\forall y\, \Phi(x,y)$ is equivalent to $\forall y\forall x\, \Phi(x,y)$ and this to $\forall x\forall y\, \Phi(y,x)$