What's the intuition behind using Law of Excluded Middle in Natural Deduction? I've recently started learning First-Order Logic and I have been doing some Natural Deduction exercises. I understand the principles behind most of the Inference Rules but when it comes to applying Classical Rules such as the law of excluded middle, I struggle to reason why it has been used.
For example:
In the proof for:
(φ → ∃x. ψ) ⊢ ∃x. (φ → ψ)


*

*φ → ∃xψ (hypothesis)

*φ ∨ ¬φ  (law of excluded middle)
...
∃x. (φ → ψ)
The solution proceeds by using law of excluded middle for φ so that you can use the ∃ elimination rule to reach the conclusion. I understand the solution but I cannot understand why someone has thought to use law of excluded middle to proceed. Is there any intuition behind this, or is it just a 'trick'?

I wouldn't call it a 'trick' as much as I would call it a good 'strategy', and over time, as you do more and more of these proofs, you'll start to recognize the times when using the Law of Excluded Middle would be good to use ... and on what statement to use it.
I also would like to point out that just because some proof works out by using the Law of Excluded Middle does not mean that you need the Law of Excluded Middle. In other words, you shouldn't feel like you wouldn't be able to complete the proof if you didn't think to use Excluded Middle.  
Indeed, in this case, you can also do a proof by Contradiction:

