Christianity is a religion which was created off of Judaism, which was a religion held by the Semites. The antisemitism is a result of new hating what they replaced. As children are angry at their parents, and so forth. Christianity originates in Judaism. They therefor hate Semites. I hope I made my point clear.
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Initially, Christianity was regarded as a sect of Judiasm, and in order to distinguish themselves from Judaism, some early Christians adopted an anti-Judaistic stance. However, they were not 'antisemites' as the term is now understood.