eqnlist/.{a_= =b_ :> (Simplify/@(a x == b x)}
or if cancellations of factors are intended
eqnlist/. {a== b :> (0==#&) /@ Factor[ Simplify[a-b]}]
And finally in order to get the logical solution set of equations
Or @@ (Solve[{a*x + b*y + c*z == 0, 1/a*(a*x + b*y + c*z) == 0, x + b/a*y + c/a*z == 0}] /. Rule -> Equal)
{z == (-a x - b y)/c} ||
{c == 0, y == -((a x)/b)} ||
{b == 0, c == 0, x == 0}
