Does the fast of one who intended to make up a missed fast at the Fajr اذان (call to prayer) and completed his fast, count as valid?

Formulating the intention for fasting at night is a condition for every obligatory fast, such as fasting in Ramadan, whether in performance or in make-up, and for vows and expiations. The proof for this is the Hadith: "Whoever does not make the intention to fast before Fajr, there is no fast for him." The intention must be made before the true dawn (Al-Fajr Al-Sadiq) appears, not just the Adhan, unless the muezzin calls the Adhan after the dawn has clearly appeared. If the muezzin relies on the calendar, then abstaining immediately upon hearing the Adhan is not obligatory; it is permissible to eat and drink until the dawn becomes clear.

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Note: Based on this, the fast is valid if there is no certainty that the muezzin is calling the Adhan precisely at the very first appearance of dawn.