I have not read this syllogism, but what I have read of Athanasius is not impressive. He doesn't think very far ahead of his words, and he does not know the scriptures very well. It is the wicked Jews that said only God can forgive sin. (Luke 5:21)
This syllogism is not reasoned out very well. When Jesus was preparing his 12 apostles to take over the church once he was gone, he gave Peter the Keys of the Kingdom of heaven, so that whatsoever he was to bind on earth, it would be bound in heaven and whatsoever he loosed on earth, it would be loosed in heaven. This has to do with the power of the apostles to forgive sins. If they forgave the sins of a man on earth, they would be forgiven in the books in heaven. (Mat. 16:19)
In Luke 5:20-23 Jesus forgave a man of his sins. The Jews said in their hearts, he blasphemes, who can forgive sins, but God only? Then Jesus said an interesting thing to them, whether is easier, to say, thy sins are forgiven thee or to say, rise up and walk? Then he said, so that you may know that the Son of Man has the power to forgive sins, he commanded the man to rise up and walk, and he was instantly healed. So which is easier to forgive sin or to heal?
In Acts 3:6 Peter was walking to the temple and stopped to behold a man asking for alms. Peter said to him money have I none, but in the name of Jesus, stand up and walk. The man stood and was instantly healed and rejoiced. Could Peter have forgiven this man his sins? Absolutely.
So Athanasius in his eagerness to prove the trinity with this syllogism, really showed his shallow knowledge of the scriptures and of priesthood power given to the apostles to save people on earth and to heal and to function in all ways in preaching the doctrines of salvation after the Savior was gone. They did a great job and billions have been saved by their good works.
We do not worship Peter and the apostles, even though they had the power to heal and forgive sins.