While I think there is some basic truth to your position, it's really not that simple. There are people who are two genetic people (called "chimeras", "fused twins", and "mosaics"). You can find an interesting article on that here. Regular twins can also create problems for your statement, as can quite a few science fiction scenarios that might not remain fiction forever.
Basically, the truth is that it was the point where you became a distinct individual but the distinction isn't entirely genetic, nor does it have to be at all. For example, at the moment an embryo breaks in two to form identical twins is the point at which each becomes a distinct individual, even though neither is genetically distinct from the other. Then there are co-joined twins, which create some interesting tests for what defines one individual or two.
On the other end of the spectrum, blood transfusions, bone marrow transplants, organ transpants, forms of cancer, and even possibly gene therapy in the future may alter the genetics of a person long after they are born, in part or in whole. So basically, genetic consistency and distinctness can be a weak hook to hang an argument on.
I'm pointing this out not to disprove that a fertilized egg is a person. I believe that, too. I've simply been in some very deep and complex abortion debates and that's the sort of left hook that pro-choice debaters are going to throw at you if you tangle with the smarter ones. Be ready to deal with them just in case.
(As a final aside, chimeras are why we also should not assume that a negative genetic test in cases like sex crimes proves that a person is innocent -- a single person can have two sets of genes.)
The safer word is, after all, not individual but person. However, there is a deep-rooted prejudice, based on Cartesian philosophy I think, that personality is simply a function of the brain, which is supposed to generate thought independently of the rest of the body. That is also simplistic, but scientists love to oversimply things just as we all do.