Praise be to Allah.
I asked our Shaykh, ‘Abd al-‘Azeez ibn ‘Abd-Allaah ibn Baaz 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 Ayyaam al-Beed, and the Bounty of Allaah is great indeed.
With regard to this particular matter, Shaykh Muhammad ibn Saalih al-‘Uthaymeen replied as follows:
“Yes, if he fasts six days of Shawwaal, he does not have to fast Ayyaam al-Beed during that month too, whether he fasted the six days at the same time as al-Ayyaam al-Beed (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. ‘Aa’ishah (may Allaah be pleased with her) said: ‘The Prophet (peace and blessings of Allaah 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.”
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… And Allaah knows best.