I'll just give this simple example to illustrate my point:
There exists a real number $x$ such that for every real number $y$, we have $xy=y$.
If I wish to prove this statement in a non-constructive way, that is to say, without explicitly assigning to $x$ the value $1$ and thereby proving the statement, I don't know how to do it.
My first problem arises in translating the sentence into logical connectives. I could say:
$$(\exists x )[x \in \mathbb{R} \, \land \forall y(y \in \mathbb{R} \implies xy=y) ]$$
But then, how could I take - for example - something like the contrapositive of the implication when it's all embedded in these quantifiers, not to mention the conjunction? Surely I can't just take the contrapositive of the embedded implication as if it were a standalone implication, considering all the intertwined connectives and quantifiers surrounding it?
Edit: Allow me to add in this quick edit. My problem is in part that I have no clue how I would go about deriving a contradiction from the negated statement above. If I wish to do a proof by contradiction, I first negate the above expression to the shorthand $\forall x \in \mathbb{R}, \exists y \in \mathbb{R}, xy \neq y$. However, what's next? I presume it would start off like:
Let $x$ be an arbitrary real number. Then...
But I don't quite understand what I would do in this particular case.