The fourth way first establishes that there is some way in which there is something most true, and it then shows why this greatest truth also exists. The proof for its existence presupposes knowledge of another text:
…there is something which is truest, something best, something noblest and consequently something which is uttermost in existence; for those things that are greatest in truth are greatest in existence, as it is written in Metaph. ii.
Notice the total reliance on the text from Metaphysics II. Though I have read many commentaries and criticisms of the Fourth Way, I have never read one that cited or discussed the relevant passage from Aristotle:
It is right also that philosophy should be called knowledge of the truth. For the end of theoretical knowledge is truth, while that of practical knowledge is action (for even if they consider how things are, practical men do not study the eternal, but what is relative and in the present). Now we do not know a truth without its cause; and a thing has a quality in a higher degree than other things if in virtue of it the similar quality belongs to the other things as well (e.g. fire is the hottest of things; for it is the cause of the heat of all other things); so that that causes derivative truths to be true is most true. Hence the principles of eternal things must be always most true (for they are not merely sometimes true, nor is there any cause of their being, but they themselves are the cause of the being of other things), so that as each thing is in respect of being, so is it in respect of truth.
There are two arguments here for why what is most true is greatest in being:
1 maj.) The eternal is a greater kind of existence than the relative, contingent, or here and now
1 min.) A something is true, so it is eternal.
So as something is true, so it has a greater existence.
The minor follows the division of speculative and practical science. The Major is self-evident ad sapientes.
The second argument
2 min.) as something is true, so it is causal
2 maj.) as something is causal, so it is existent.
So as something is true, so it is existent.
The minor is clear from the axioms and principles of the sciences, which must be more true than the conclusions (see APo book 1); and the major is clear from causality being an operation which follows the existence of something.