Proving symmetry algebraically about origin and axes.

Hello!

how could i show that any graph that is symmetrical to the origin as well as the y-axis must also be symmetrical to the x-axis?

i understand how to determine if a given function is symmetrical to the origin, x-axis, and y-axis

intuitively i believe that the statement in the above post is true. but i have no idea how to prove this.

edit: ok, so i've thought about it a bit. i'm thinking you would first pick a point on the graph, lets say in QI. say it is (2,1). then rotate it 180 degrees about the origin giving a point of (-2,-1). starting with that same point again, (2,1) i would know that another point exists on the graph that is (-2,1) or the y-axis symmetrical point. i see that the points created with the point symmetry of the origin and the line symmetry of the y-axis are symmetrical across the x-axis.

however, i could draw a graph that goes through each of these three points and does not go through (2,-1). a point that is important to show x-axis symmetry.

could someone please explain to me how to prove this algebraically? or prove that it is not true algebraically?

thank you!