I guess something like this would return true or false, depending on whether the implication holds true or not. Is there a way to do this?
There are utilities for manipulating equations. Using them in sequence result in a proof:
x == 2 ApplySides[#^2 &, %]
This gives the result without proof:
Reduce[x == 2 \[Implies] x*x == 4]
If you try
Simplify[p,q]
Mathematica will try to simplify p given the information supplied in q.
For your example
Simplify[x*x==4,x==2]
returns
True
See if that will do the kinds of things you are looking for.
