On a re-reading of D.J.Vellemann's book - "How to prove it" (2nd edition, pg. 69), it reads
It should be clear that if $A =\varnothing$, then $\exists x \in A P(x)$ will be false no matter what the statement $P(x)$ is. There can be nothing in $A$ that, when plugged in for $x$, makes $P(x)$ come out true as well.
This second line actually means that suppose $P(x)$ is true, then we can't find anything that proves it to be true. In other words, one cannot prove that $P(x)$ is true rather than proving $P(x)$ is false. Am I right in this reasoning?