Subhomomorphism relation between transfer condition operator and intersection operator

Statement
Suppose $$\cap$$ denotes the intersection operator on subgroup metaproperties and $$T$$ denotes the fact about::transfer condition operator. Then, we have:

$$\! T(p) \cap T(q) \le T(p \cap q)$$.

Related facts

 * Transfer condition operator preserves intersection-closedness
 * Intersection-closure operator preserves transfer condition