1. Something exists. (Premise)
2. Something exists if and only if being exists. (Premise)
3. Therefore, being exists. (From 1 and 2)
I use the expression "being exists" in order to avoid the redundancy of saying that "existence exists." Nevertheless, "being" is used synonymously with "existence."
Premise (1) is quite obviously true. For example, in order to doubt my own existence, I would first have to exist in order to doubt it. This is along the same lines as Descartes's cogito ergo sum, or "I think, therefore I am."
Premise (2) is more controversial. However, I'm not at all impressed by the objections to it. Kant asserted that existence is not a predicate. But, why should we agree with such a contention? What, for instance, is the difference between something real and something not-real? If being does not exist, then the difference between something real and something not-real is literally nothing. This means that what is real is identical to what is not-real, which is absurd.
Thomas continues by deducing some of the divine attributes of being, or "being itself subsisting." Being must be eternal and omnipresent, since there is no time or place at which being does not exist. If there were no being at some time or place, then nothing would exist at either time or place. Moreover, being must be unique, or one. If there were more than one being, then there would be distinctions among them. However, to be distinct from being is to be non-being, in which case the latter does not exist anyway. Being is a thing's existence or actuality. The reason why things are able to be distinct from one another is because they possess distinct essences. The essence of a thing is its nature. This means that everything that exists participates in one being, but can be distinguished by their various essences.
Being must also be immutable, since to change from being to anything else would result in being's becoming non-being, which is self-contradictory. Next, being must contain within itself all perfections: omnipotence, omniscience, and moral perfection. If it did not possess these attributes, then being would exhibit potentiality, which is impossible for something immutable. Changing things exhibit actuality and potentiality, just as the acorn is merely an acorn in actuality and an oak tree in potentiality. Being itself must be Pure Actuality as a result of its immutability.
Therefore, I maintain we have a sound and rationally compelling argument for the existence of an eternal, omnipresent, unique, immutable, omnipotent, omniscient, and morally perfect being. This, as the Angelic Doctor muses, everyone understands to be God.