An easement of necessity might arise if your only way to the public road was through your neighbour's property. As I understand it, neither of you have access to the public road except by building a bridge. So an easement of necessity probably does not arise - you and he are both equally free to build bridges.
An easement generally is over real property, i.e. land and its fixtures. The bridge is probably a fixture to the land, and it might be the subject of an easement. But the question you are really faced with is, what happens now that the bridge is destroyed? Is your neighbour obliged to repair it? I am not exactly sure of the answer, but by analogy to contract law and the doctrine of frustration, I would suspect the answer is "no". If you and he contracted to allow you to use the bridge, and the bridge was destroyed, it would be impossible to fulfill the contract. The contract would be "frustrated", and would be automatically discharged. I suspect the same is true here - you have a right to use the bridge while it exists, but he is under no obligation to rebuild the bridge if it is destroyed.
So what you need to do, legally, is establish not only that you have the right to use the bridge - which is pretty empty right now - but establish that he is obliged to repair it. Frankly, I think that is unlikely under the general principles of property law. I don't know the specific laws of your jurisdiction, and I suppose there might be some sort of reliance or estoppel argument you could make, so I could be wrong about that. You should talk to a local real estate lawyer to be certain.
Prior to doing that, I would explore your options with your neighbour. Why is it he wants separate bridges? Has the previous arrangement been unsatisfactory? Is there any way of building one bridge on his property and allowing you access to it?