What's the reason for this symmetry argument?
If you reversed the battery you would have the same circuit. You could see this by reversing the battery, and then redrawing the circuit horizontally flipped - you would get the exact same circuit, but with different labels.

So FB in the reversed circuit would be in the same position as AC in the original circuit.

Since your circuit only has resistors, it has symmetry regarding the direction of current flow. Reversing the battery doesn't change the amount of current flowing through each resistor - it only reverses the direction of the current flow.

You should be able to see that the current flowing out of AC in the reversed circuit is the same as the current flowing into AC in the original circuit, because only the current direction is changed. But I also just showed you that AC in the reversed circuit is the same as FB in the original circuit. Therefore, the amount of current flow through AC in the original circuit, and FB in the original circuit, is the same.