Let G be a group and H,K subgroups of G. If (x,y both belong to G ), define a relation ~ on G by x~y if y=axb for some (a in H) and (b in K). Prove that ~ is an equivalence relation

I tried to solve the question and this is what I was able to say so far:

I know to show ~ equivalent relation it has to be reflexive, symmetric, and transitive

I know for reflexive {for all u in G u~u}

for symmetric {for all u, v in G whenever u~v then v~u}

for transitive { for all u, v, w in G whenever u~v and v~w then u~w}. Here I need to start by " Let u~v and v~w for all u, v, w in G and I want to show u~w"