Why is this true? $(\exists x)(P(x) \Rightarrow (\forall y) P(y))$ Why is this true?
$(\exists x)(P(x) \Rightarrow (\forall y) P(y))$

In mathematics, if not in philosophy, the domain of quantification is usually made explicit in statements. If we let the domain be some set $U$, your statement becomes 
$$\exists x\in U (P(x) \Rightarrow \forall y\in U (P(y))$$ 
or equivalently
$$\exists x(x\in U \wedge (P(x) \Rightarrow \forall y(y\in U\Rightarrow P(y))))$$ 
Then you can formally prove this statement is true iff $U$ is non-empty as sketched here: 
For non-empty $U$, consider two cases: 
$$\forall y\in U (P(y))\lor \neg\forall y\in U (P(y))$$ 
In each case, obtain
$$\exists x\in U (P(x) \Rightarrow \forall y\in U (P(y))$$ 
For empty U, assume to the contrary that 
$$\exists x\in U (P(x) \Rightarrow \forall y\in U (P(y))$$ 
and obtain a contradiction.
Full text of formal proof at http://dcproof.com/Mats.htm
