I think this question deserves a formal answer.

Let;s work a slightly more complicated example.

eqns = {x^2 + y^2 == 1, -x^2 - y^2 + 1 == 0, 5 x - 1 == y, y == 5 x - 1};
eqns // Simplify // DeleteDuplicates

{x^2 + y^2 == 1, 5 x == 1 + y}

The idea is to regularize the equations and delete the duplicates.