If you are not concerned about your costs, then generally, yes you can tear it down as long as you immediately replace it with a fence of equal or greater value. What fully constitutes value may depend upon the situation there. Factors beyond material costs to consider may be views, containment ability, durability. There may be other factors. Consider what the current fence is used for to make these judgements.
If you are concerned about being reimbursed for the costs your neighbor rightly should pay, as Mr. Knowitall said, you will likely only be entitled to 1/2 of the cost of labor and materials to replace the existing fence with like materials, or if the fence is repairable at a lesser cost, 1/2 of that cost. Anything above that is on you unless the neighbor then upgrades all her other fences to match your new one. That seems unlikely in the situation you describe.
If you do not already have surveyed boundaries, I suggest getting a survey done to ensure that your new fence is placed on the correct line. If you have not had a survey but are satisfied with the location of the existing fence, simply put the new one in the exact same location. However, without having a surveyed line, you risk having to live with this line even if at some point in the future you find by survey that the true line was elsewhere.

