In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
Responding to the question, Sheikh M. S. Al-Munajjid, a prominent Saudi Muslim lecturer and author, states:
I asked our Sheikh Ibn Baz about this matter, and he replied that hopefully he will be rewarded for that, because it is true that he fasted the six days, and it is also true that he fasted the white days, and the Bounty of Allah is great indeed.
With regard to this particular matter, Sheikh Ibn Uthaymeen replied as follows:
“Yes, if he fasts six days of Shawwal, he does not have to fast white days during that month too, whether he fasted the six days at the same time as the white days (the 13th, 14th and 15th of the Hijri month) or before them or after them, because it is true that he has fasted three days of the month.
Aishah (may Allah be pleased with her) said: ‘The Prophet (peace be upon him) used to fast three days of every month, and he did not bother whether he fasted them at the beginning of the month or in the middle or at the end.”
📚 Read Also: Why Are We Encouraged to Fast Six Days of Shawwal?
This case is similar to that of tahiyyat al-Masjid (prayer to “greet the mosque” upon entering), which does not have to be done if one prays a regular prayer upon entering the mosque. So if you enter the mosque and pray a regular Sunnah prayer, you do not have to pray tahiyyat al-masjid.
Almighty Allah knows best.