Peter and Paul were martyred in Rome

We read the lives of the Cæsars: At Rome Nero was the first who stained with blood the rising faith. Then is Peter girt by another, when he is made fast to the cross. Then does Paul obtain a birth suited to Roman citizenship, when in Rome he springs to life again ennobled by martyrdom.  Wherever I read of these occurrences, so soon as I do so, I learn to suffer; nor does it signify to me which I follow as teachers of martyrdom, whether the declarations or the deaths of the apostles, save that in their deaths I recall their declarations also.

Tertullian
Scorpiace
Chapter XV

The Lord gave the keys to heaven to Peter, then to the Church, then to every believer

For though you think heaven still shut, remember that the Lord left here to Peter and through him to the Church, the keys of it, which every one who has been here put to the question, and also made confession, will carry with him.

Tertullian
Scorpiace
Chapter X

The revelation of the Trinity makes the New Covenant possible

What need would there be of the gospel, which is the substance of the New Covenant, laying down (as it does) that the Law and the Prophets lasted until John the Baptist, if thenceforward the Father, the Son, and the Spirit are not both believed in as Three, and as making One Only God? God was pleased to renew His covenant with man in such a way as that His Unity might be believed in, after a new manner, through the Son and the Spirit, in order that God might now be known openly, in His proper Names and Persons, who in ancient times was not plainly understood, though declared through the Son and the Spirit.

Tertullian
Against Praxeas
Chapter XXXI

The Holy Spirit is the third person of the Godhead

He will come again on the clouds of heaven, just as He appeared when He ascended into heaven. Meanwhile He has received from the Father the promised gift, and has shed it forth, even the Holy Spirit—the Third Name in the Godhead, and the Third Degree of the Divine Majesty; the Declarer of the One Monarchy of God…

Tertullian
Against Praxeas
Chapter XXX

The Father did not suffer

Then, again, the Father is as incapable of fellow-suffering as the Son even is of suffering under the conditions of His existence as God. Well, but how could the Son suffer, if the Father did not suffer with Him? My answer is, The Father is separate from the Son, though not from Him as God.

Tertullian
Against Praxeas
Chapter XXIX

Jesus is the Word of God and man

But the truth is, we find that He is expressly set forth as both God and Man; the very psalm which we have quoted intimating (of the flesh), that “God became Man in the midst of it, He therefore established it by the will of the Father,”—certainly in all respects as the Son of God and the Son of Man, being God and Man, differing no doubt according to each substance in its own especial property, inasmuch as the Word is nothing else but God, and the flesh nothing else but Man. Thus does the apostle also teach respecting His two substances, saying, “who was made of the seed of David;” in which words He will be Man and Son of Man.  “Who was declared to be the Son of God, according to the Spirit;” in which words He will be God, and the Word—the Son of God. We see plainly the twofold state, which is not confounded, but conjoined in One Person—Jesus, God and Man.

…In like manner, again, the apostle calls Him “the Mediator between God and Men,” and so affirmed His participation of both substances.

Tertullian
Against Praxeas
Chapter XXVII

Christians are baptised by immersion into three persons

After His resurrection He promises in a pledge to His disciples that He will send them the promise of His Father; and lastly, He commands them to baptize into the Father and the Son and the Holy Ghost, not into a unipersonal God. And indeed it is not once only, but three times, that we are immersed into the Three Persons, at each several mention of Their names.

Tertullian
Against Praxeas
Chapter XXVI

The Son and Spirit belong to the Father and so are not the same as the Father, but are both God

For both the Spirit is the substance of the Word, and the Word is the operation of the Spirit, and the Two are One (and the same). Now John must mean One when he speaks of Him as “having been made flesh,” and the angel Another when he announces Him as “about to be born,” if the Spirit is not the Word, and the Word the Spirit. For just as the Word of God is not actually He whose Word He is, so also the Spirit (although He is called God) is not actually He whose Spirit He is said to be. Nothing which belongs to something else is actually the very same thing as that to which it belongs.

…And thus the Spirit is God, and the Word is God, because proceeding from God, but yet is not actually the very same as He from whom He proceeds. Now that which is God of God, although He is an actually existing thing, yet He cannot be God Himself (exclusively), but so far God as He is of the same substance as God Himself, and as being an actually existing thing

Tertullian
Against Praxeas
Chapter XXVI

Father, Son and Spirit are three persons but one essence

Thus the connection of the Father in the Son, and of the Son in the Paraclete, produces three coherent Persons, who are yet distinct One from Another. These Three are one essence, not one Person, as it is said, “I and my Father are One,” in respect of unity of substance not singularity of number.

Tertullian
Against Praxeas
Chapter XXV

John 1:1 can only be interpreted to mean one thing — Christ is God but distinct from the Father

…first of all there comes at once to hand the preamble of John to his Gospel, which shows us what He previously was who had to become flesh.  “In the beginning was the Word, and the Word was with God, and the Word was God. He was in the beginning with God: all things were made by Him, and without Him was nothing made.” Now, since these words may not be taken otherwise than as they are written, there is without doubt shown to be One who was from the beginning, and also One with whom He always was: one the Word of God, the other God (although the Word is also God, but God regarded as the Son of God, not as the Father); One through whom were all things, Another by whom were all things.  But in what sense we call Him Another we have already often described. In that we called Him Another, we must needs imply that He is not identical—not identical indeed, yet not as if separate; Other by dispensation, not by division.

Tertullian
Against Praxeas
Chapter XXI