It works, but has more steps than necessary.
But it does work.
Q:Prove that P(A∩B)=P(A)∩P(B).
A:Let x∈P(A∩B) be arbitrary ⇒ x⊆A∩B
⇔ ∀y(y∈x⇒y∈A and y∈B)
⇔ x⊆A and x⊆B
⇔ x∈P(A) and x∈P(B)
⇔ x∈P(A)∩P(B) ■
Does this work? I'm not sure if I have used the logical connectives correctly, such as breaking up the subset definition the way I did. I'm just trying prove things using iff statements for the practice.