Does an equivalence still hold when quantified?

Question

I'm trying to prove $\vdash(\exists x)((A\land B)\land C))\equiv(\exists x)((A\land(B\land C))$. In propositional logic, I proved that conjunction is associative. So the fact that these formulas are now quantified, does the associativity still hold? For that matter, what of $\vdash(\forall x)((A\land B)\land C))\equiv (\forall x)((A\land(B\land C))$? That is, can I justify such proofs by simply noting that conjunction is associative, or is it more complicated than that.

Answer 1

Both statements are tautologies solely by the associativity of conjunction. If for a given $x$ $A\land(B\land C)$ is true, then $(A\land B)\land C$ is true; if the former statement holds for all $x$, so does the latter.