Photius used this logic: if God’s common attributes belong only to His ousia, then spirating, common to Father and Son, must belong to ousia. But since the Spirit spirates no one, the Spirit has a different ousia and is not God. (Photius, Mystagogy 34).
Consubstantiality is the source of the Catholic teaching (from the Symbol of Faith). The Father and the Son and the Holy Spirit, all participate in the spiration.
St. Augustine: “As the Father has it in himself that the Holy Spirit should proceed from him, so has he given it to the Son that the same Holy Spirit should proceed from him, without reference to time in the two cases. And it is said that the Holy Spirit proceeds from the Father, in such a way that it may be understood that, if he proceeds also from the Son, this is something the Son possesses from the Father. In fact, whatever the Son has, he has from the Father; he has from the Father that the Holy Spirit proceed from him…. The Son is born of the Father; and the Holy Spirit proceeds from the Father principally (principaliter) and, by the intemporal gift of this to the Son, from the Father and the Son, in communion (communiter).
From: De Trinitate, liber XV, cap. 25.47, PL 42, 1094-1095
Lyons II (1274):
“We believe also that the Holy Spirit is complete and perfect and true God, proceeding from the Father and the Son, coequal and consubstantial, co-omnipotent, and coeternal through all things with the Father and the Son. We believe that this Holy Trinity is not three Gods but one God, omnipotent, eternal, invisible, and unchangeable.” … “condemn and reject who presume to deny that the Holy Spirit proceeds eternally from the Father and the Son; as well as who with rash boldness presume to declare that the Holy Spirit proceeds from the Father and the Son as from two beginnings, and not as from one.”
Council of Florence (1429):
“And since all that the Father has, the Father Himself, in begetting, has given to His Only-Begotten Son, with the exception of Fatherhood, the very fact that the Holy Spirit proceeds from the Son, the Son Himself has from the Father eternally, by Whom He was begotten also eternally.”
Pope Gregory XIII (1575):
“All things, however, which are of the Father, the Father Himself has given to His Only-Begotten Son in generation, outside of being the Father; the very fact that the Holy Spirit proceeds from the Son, the Son Himself eternally has from the Father, by Whom He has also been eternally begotten.”