Bear with me in this post, I am no means a scholar in Theology or Philosophy, but I do believe this dogma not defined regarding the Office of the Papacy was touched on by Henry Edward Cardinal Manning first in the pages 55 and 56 of his book The Vatican Council And Its Definitions: A Pastoral Letter To The Clergy
"For the Church, which, together with the Apostolic office of teaching, received also the command to guard the deposit of faith is divinely invested with the right and duty of proscribing science falsely so-called, lest any man be deceived by philosophy and vain deceit." Henry Edward Cardinal Manning. Page 55.
"This truth has been traditionally held and taught by the Holy See, by the praxis of the Church, and by the Ecumenical Councils, especially those in which the East and the West met in union together, as for instance the fourth of Constantinople, the second of Lyons, and the Council of Florence. It is then declared, that in virtue of the promise of Our Lord, "I have prayed for thee, that by thy faith fail not," a perpetual grace of stability in faith was divinely attached to Peter and to his successors in his See." Henry Edward Cardinal Manning, Page 62. Now, this dogma being defined by Cardinal Manning has pretty much been by Sacred Scripture and other previous councils, a true question must be asked, has this perpetual grace, is this seen in the man chosen by the College of Cardinals, Jorge Bergoglio? Has he denounced bishops and bishop councils from teaching novelties and errant philosophies and heresies? What about the German Bishops approving homosexuality as normal? The Argentinian Bishops giving Holy Communion to Divorced & Civilly Remarried couples? That heretical document Amoris Latietia? Pagan Light Shows on the Churches of Rome? Allowing Pagan Worship in the Vatican and in Saint Peter's Basilica? Saying we should support the United Nations? Placing Abortion/Population Control Experts on the Pontifical Academy of Life? Protecting Sexual Abusing Priests, Keeping A Homosexual Priest as a Priest in Netherlands? Slapping a faithful Catholic from China and calling her either a witch or a worse curse word?
Do we need more from Henry Edward Cardinal Manning? Let us see what else he says. This at the end of Page 64, he begins to quote on something called cathedra Petri
"which in St. Cyprian, St. Optatus, and St. Augustine is employed as synonymous with the successor of Peter, and is used to express the centre and test of Catholic unity."
As you can see, Papal Infallibility and Fidelity To The Office Of The Papacy, go hand and hand. Without one or the other, the Holy Father could not governor the Church, defend the Faith, and condemn errant clergy and other abominations.
I must make one final note, this is not in defense of the Protestants in Catholic clothing who call themselves Sedevacantists, but in the defense of the Papal Office and the Authority of the Pope, given to Saint Peter by Our Lord, to show who has been given down through the line of successors of Saint Peter to the current Pope, and to see and to make sure the current man has been given this grace.
To read more of this book I quoted from, it is free to read and download, The Vatican Council and Its Definitions: A Pastoral Letter by Henry Edward Cardinal Manning