Talmud Bavli, Beitzah
Beitzah 4b
The Talmud discusses why the second day of Yom Tov in the Diaspora was retained even after the calendar became fixed, concluding that the rabbis maintained the custom because 'be careful with the custom of your ancestors' — the practice of the exiles is a decree that may not be uprooted.
אִתְּמַר: שְׁנֵי יָמִים טוֹבִים שֶׁל גָּלִיּוֹת, רַב אָמַר: נוֹלְדָה בָּזֶה — מוּתֶּרֶת בָּזֶה, וְרַב אַסִּי אָמַר: נוֹלְדָה בָּזֶה — אֲסוּרָה בָּזֶה. וְהַשְׁתָּא דְּיָדְעִינַן בִּקְבִיעָא דְיַרְחָא, מַאי טַעְמָא עָבְדִינַן תְּרֵי יוֹמֵי? מִשּׁוּם דִּשְׁלַחוּ מִתָּם: הִזָּהֲרוּ בְּמִנְהַג אֲבוֹתֵיכֶם בִּידֵיכֶם, זִמְנִין דְּגָזְרוּ הַמַּלְכוּת גְּזֵרָה וְאָתֵי לְאִקַּלְקוּלֵי.
§ It was stated that there is a dispute between amora’im with regard to the halakha for the two Festival days observed in the Diaspora. Rav said: An egg that was laid on this day is permitted on that one, and Rav Asi said: An egg that was laid on this day is prohibited on that one. The Gemara asks: And now that we know the determination of the first day of the new month, what is the reason that we observe two Festival days in the Diaspora? Because they sent a warning from there, from Eretz Yisrael: Although now there is a fixed calendar and there is no uncertainty, be careful to observe the custom of your fathers that you received, because at times the monarchy will issue decrees of persecution restricting Torah study and the fixed calendar may be forgotten. And the people will come to have their proper observance of the Festivals be disrupted again. However, the fundamental halakha is that the observance of two Festival days is based on uncertainty.
Why it matters — Classic Talmudic source establishing that Yom Tov Sheni in the Diaspora was preserved as a rabbinic decree even after its original rationale (doubt about the calendar) no longer applied.