Christ is distinct in personhood from the Father, but not substance

… it was because He had already His Son close at His side, as a second Person, His own Word, and a third Person also, the Spirit in the Word, that He purposely adopted the plural phrase, “Let us make;” and, “in our image;” and, “become as one of us.” For with whom did He make man? and to whom did He make him like? (The answer must be), the Son on the one hand, who was one day to put on human nature; and the Spirit on the other, who was to sanctify man. With these did He then speak, in the Unity of the Trinity, as with His ministers and witnesses.

But all the rest of the created things did He in like manner make, who made the former ones—I mean the Word of God, “through whom all things were made, and without whom nothing was made.” Now if He too is God, according to John, (who says,) “The Word was God,” then you have two Beings—One that commands that the thing be made, and the Other that executes the order and creates. In what sense, however, you ought to understand Him to be another, I have already explained, on the ground of Personality, not of Substance—in the way of distinction, not of division. But although I must everywhere hold one only substance in three coherent and inseparable (Persons), yet I am bound to acknowledge, from the necessity of the case, that He who issues a command is different from Him who executes it.

Tertullian
Against Praxeas
Chapter XII

The Father and the Son are distinct from each other because they speak to each other

On your side, however, you must make Him out to be a liar, and an impostor, and a tamperer with His word, if, when He was Himself a Son to Himself, He assigned the part of His Son to be played by another, when all the Scriptures attest the clear existence of, and distinction in (the Persons of) the Trinity, and indeed furnish us with our Rule of faith, that He who speaks, and He of whom He speaks, and to whom He speaks, cannot possibly seem to be One and the Same.

Still, in these few quotations the distinction of Persons in the Trinity is clearly set forth. For there is the Spirit Himself who speaks, and the Father to whom He speaks, and the Son of whom He speaks. In the same manner, the other passages also establish each one of several Persons in His special character—addressed as they in some cases are to the Father or to the Son respecting the Son, in other cases to the Son or to the Father concerning the Father, and again in other instances to the (Holy) Spirit.

Tertullian
Against Praxeas
Chapter XI

The Father, Son and Holy Spirit are one and yet three

Bear always in mind that this is the rule of faith which I profess; by it I testify that the Father, and the Son, and the Spirit are inseparable from each other, and so will you know in what sense this is said. Now, observe, my assertion is that the Father is one, and the Son one, and the Spirit one, and that They are distinct from Each Other.

Tertullian
Against Praxeas
Chapter IX

The Logos is numerically distinct from the Father but as a fountain from a river they are also one

“For what man knoweth the things which be in God, but the Spirit which is in Him?” But the Word was formed by the Spirit, and (if I may so express myself) the Spirit is the body of the Word. The Word, therefore, is both always in the Father, as He says, “I am in the Father;” and is always with God, according to what is written, “And the Word was with God;” and never separate from the Father, or other than the Father, since “I and the Father are one.”

…Following, therefore, the form of these analogies, I confess that I call God and His Word—the Father and His Son—two. For the root and the tree are distinctly two things, but correlatively joined; the fountain and the river are also two forms, but indivisible; so likewise the sun and the ray are two forms, but coherent ones. Everything which proceeds from something else must needs be second to that from which it proceeds, without being on that account separated.  Where, however, there is a second, there must be two; and where there is a third, there must be three. Now the Spirit indeed is third from God and the Son; just as the fruit of the tree is third from the root, or as the stream out of the river is third from the fountain, or as the apex of the ray is third from the sun. Nothing, however, is alien from that original source whence it derives its own properties.  In like manner the Trinity, flowing down from the Father through intertwined and connected steps, does not at all disturb the Monarchy, whilst it at the same time guards the state of the Economy.

Tertullian
Against Praxeas
Chapter VIII

The Logos was begotten as God said “Let there be light”

Then, therefore, does the Word also Himself assume His own form and glorious garb, His own sound and vocal utterance, when God says, “Let there be light.” This is the perfect nativity of the Word, when He proceeds forth from God—formed by Him first to devise and think out all things under the name of Wisdom—“The Lord created or formed me as the beginning of His ways;” then afterward begotten, to carry all into effect—“When He prepared the heaven, I was present with Him.” Thus does He make Him equal to Him: for by proceeding from Himself He became His first-begotten Son, because begotten before all things; and His only-begotten also, because alone begotten of God, in a way peculiar to Himself, from the womb of His own heart—even as the Father Himself testifies: “My heart,” says He, “hath emitted my most excellent Word.”

Tertullian
Against Praxeas
Chapter VII

God the Father was never alone because His Word was always within Him.

For before all things God was alone—being in Himself and for Himself universe, and space, and all things. Moreover, He was alone, because there was nothing external to Him but Himself.  Yet even not then was He alone; for He had with Him that which He possessed in Himself, that is to say, His own Reason. For God is rational, and Reason was first in Him; and so all things were from Himself.  This Reason is His own Thought (or Consciousness) which the Greeks call λόγος, by which term we also designate Word or Discourse and therefore it is now usual with our people, owing to the mere simple interpretation of the term, to say that the Word was in the beginning with God; although it would be more suitable to regard Reason as the more ancient; because God had not Word from the beginning, but He had Reason even before the beginning; because also Word itself consists of Reason, which it thus proves to have been the prior existence as being its own substance. Not that this distinction is of any practical moment. For although God had not yet sent out His Word, He still had Him within Himself, both in company with and included within His very Reason, as He silently planned and arranged within Himself everything which He was afterwards about to utter through His Word. Now, whilst He was thus planning and arranging with His own Reason, He was actually causing that to become Word which He was dealing with in the way of Word or Discourse. And that you may the more readily understand this, consider first of all, from your own self, who are made “in the image and likeness of God,” for what purpose it is that you also possess reason in yourself, who are a rational creature, as being not only made by a rational Artificer, but actually animated out of His substance. Observe, then, that when you are silently conversing with yourself, this very process is carried on within you by your reason, which meets you with a word at every movement of your thought, at every impulse of your conception. Whatever you think, there is a word; whatever you conceive, there is reason.  You must needs speak it in your mind; and while you are speaking, you admit speech as an interlocutor with you, involved in which there is this very reason, whereby, while in thought you are holding converse with your word, you are (by reciprocal action) producing thought by means of that converse with your word. Thus, in a certain sense, the word is a second person within you, through which in thinking you utter speech, and through which also, (by reciprocity of process,) in uttering speech you generate thought. The word is itself a different thing from yourself. Now how much more fully is all this transacted in God, whose image and likeness even you are regarded as being, inasmuch as He has reason within Himself even while He is silent, and involved in that Reason His Word!

Tertullian
Against Praxeas
Chapter V

The Son is of the same substance and the Spirit proceeds from the Father through the Son

But as for me, who derive the Son from no other source but from the substance of the Father, and (represent Him) as doing nothing without the Father’s will, and as having received all power from the Father, how can I be possibly destroying the Monarchy from the faith, when I preserve it in the Son just as it was committed to Him by the Father? The same remark (I wish also to be formally) made by me with respect to the third degree in the Godhead, because I believe the Spirit to proceed from no other source than from the Father through the Son. Look to it then, that it be not you rather who are destroying the Monarchy, when you overthrow the arrangement and dispensation of it, which has been constituted in just as many names as it has pleased God to employ.

Tertullian
Against Praxeas
Chapter IV

The Son and the Holy Spirit are the same substance as the Father

Therefore, inasmuch as the Divine Monarchy also is administered by so many legions and hosts of angels, according as it is written, “Thousand thousands ministered unto Him, and ten thousand times ten thousand stood before Him;” and since it has not from this circumstance ceased to be the rule of one (so as no longer to be a monarchy), because it is administered by so many thousands of powers; how comes it to pass that God should be thought to suffer division and severance in the Son and in the Holy Ghost, who have the second and the third places assigned to them, and who are so closely joined with the Father in His substance, when He suffers no such (division and severance) in the multitude of so many angels? Do you really suppose that Those, who are naturally members of the Father’s own substance, pledges of His love, instruments of His might, nay, His power itself and the entire system of His monarchy, are the overthrow and destruction thereof? You are not right in so thinking.

Tertullian
Against Praxeas
Chapter III

The pagans struggle believing in one God, and see the three persons as three gods

The simple, indeed, (I will not call them unwise and unlearned,) who always constitute the majority of believers, are startled at the dispensation (of the Three in One), on the ground that their very rule of faith withdraws them from the world’s plurality of gods to the one only true God; not understanding that, although He is the one only God, He must yet be believed in with His own οἰκονομία . The numerical order and distribution of the Trinity they assume to be a division of the Unity; whereas the Unity which derives the Trinity out of its own self is so far from being destroyed, that it is actually supported by it.

Tertullian
Against Praxeas
Chapter III

The Trinity is one God in substance but three persons

But keeping this prescriptive rule inviolate, still some opportunity must be given for reviewing (the statements of heretics), with a view to the instruction and protection of divers persons; were it only that it may not seem that each perversion of the truth is condemned without examination, and simply prejudged; especially in the case of this heresy, which supposes itself to possess the pure truth, in thinking that one cannot believe in One Only God in any other way than by saying that the Father, the Son, and the Holy Ghost are the very selfsame Person. As if in this way also one were not All, in that All are of One, by unity (that is) of substance; while the mystery of the dispensation is still guarded, which distributes the Unity into a Trinity, placing in their order the three Persons—the Father, the Son, and the Holy Ghost: three, however, not in condition, but in degree; not in substance, but in form; not in power, but in aspect; yet of one substance, and of one condition, and of one power, inasmuch as He is one God, from whom these degrees and forms and aspects are reckoned, under the name of the Father, and of the Son, and of the Holy Ghost. How they are susceptible of number without division, will be shown as our treatise proceeds.

Tertullian
Against Praxeas
Chapter II