Mathematica 10.3 gets it wrong with this calculation (A [Intersection] B) [Union] (A [Intersection] C) It seems to work if we inactivate Union and Intersection:

Inactivate[
 replRule := {HoldPattern[
     x \[Element] 
      u_ \[Union] v_] :> (x \[Element] u) \[Or] (x \[Element] v), 
   HoldPattern[
     x \[Element] 
      u_ \[Intersection] v_] :> (x \[Element] u) \[And] (x \[Element] 
       v)};
 predicate1 = 
  x \[Element] (A \[Intersection] B) \[Union] (A \[Intersection] C);
 predicate2 = x \[Element] A && (x \[Element] B || x \[Element] C);
 Simplify[Equivalent[predicate1, predicate2] //. replRule], 
 Union | Intersection | C]

How about replacement rules:

HoldForm[x \[Element] (A \[Intersection] 
      B) \[Union] (A \[Intersection] C)] //. {HoldPattern[
    x \[Element] 
     u_ \[Union] v_] :> (x \[Element] u) \[Or] (x \[Element] v), 
  HoldPattern[
    x \[Element] 
     u_ \[Intersection] v_] :> (x \[Element] u) \[And] (x \[Element] 
      v)}