While equivalences are very useful, we are often interested in implications such as the one mentioned previously: ∀x :(φ) ⇒∃x :(φ). We could rephrase that as an equivalence, ∀x :(φ) ⇒∃x :(φ) ≡ true. Informally, it should be clear that that is rather awkward, and formally it is as well.
But such implications are exactly what inference rules are good for. So, let's continue and consider what First-order inference rules should be.