Have a hard time to conclude when a given predicate logic formula is invalid/valid. For example, this is one that I have spent a lot of time on
$$\vdash \exists x P(x) \wedge \exists x (P(x) \rightarrow Q(x)) \rightarrow \exists x Q(x)$$
To start with I have just looked at it but it makes no sense at all. So I tried to decompose it into words
If for some P and if some P result in some Q is true, then some Q is true.
But it does not really make it easier. How should I tackle a problem like this? If I conclude it is valid I "just" perform natural deduction, but if it is invalid (as I expect), how should I start constructing a counter example when I do not really understand the formula itself?