The Catholic doctrine on Christ and His Mystical Body as necessary for salvation is a truth that is unchangeable because it is Catholic dogma, in other words it is a truth declared by the Church to be revealed by God and transmitted from the Apostles in Tradition or the Bible.
It has been constantly taught, for 2,000 years, in the scriptures and by numerous councils, popes, and synods, most explicitly in the following documents: Firmiter of the Fourth Lateran Council (1215), Unam Sanctam (1302), the Decree for the Jacobites (1442), Singulari Quadam (1854), Quanto Conficiamur Moerore (1863), Mystici Corporis Christi (1943), Suprema Haec Sacra (1949), Humani Generis (1950), Lumen Gentium of Vatican II (1965), and Dominus Iesus (2000).
See also:
Joseph Fenton, The Catholic Church and Salvation (Round Top, New York: Seminary Press, 2006)
Ralph Martin, Will Many Be Saved?: What Vatican II Actually Teaches and Its Implications for the New Evangelization (Grand Rapids, MI: Eerdmans, 2012). Endorsed by Cardinals Francis George, Peter Turkson, Timothy Dolan, Donald Wuerl; Archbishop Augustine Di Noia, Archbishop Robert Carlson, Bishop David Ricken.