That's a myth.
The original emphasis was on Epiphany, not Christmas. Only later during the Victorian era did Christmas take centerstage. And the placement of Epiphany, to my knowledge, was unrelated to Saturnalia (though Saturnalia influences western secular Christmas traditions, it did not create the actual religious holiday). It was related to Hanukkah, the festival of Lights. Just as Hanukkah celebrates the Maccabean restoration of the temple and the miracle of light that occurred there, Epiphany celebrates Christ as the Light of the World, the glory of the presence of God returning to his people in the temple. In this case, Christ himself is the temple in his flesh.
In the early Church, Epiphany and Pascha/Easter were the major holidays, and this is still emphasized in the Eastern Church, with the Nativity and Passion being signs of the humility of Christ in his Incarnation and Passion, followed by his glorification in Epiphany/Theophany and the Resurrection.
I think it is interesting that Ishtar and Tammuz both contributed their names to the calendars of Christianity and Judaism, considering that they are the Elist equivalents of Mary and Jesus.
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