Are all men, then, saved by Christ just as they perished through Adam?

No. Only those are saved who by a true faith are grafted into Christ and accept all His benefits.

Scripture proofs King James Version

Matt. 7:14 Because strait is the gate, and narrow is the way, which leadeth unto life, and few there be that find it.
John 1:12 But as many as received him, to them gave he power to become the sons of God, even to them that believe on his name:
John 3:16, 18, 36 For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life. ... He that believeth on him is not condemned: but he that believeth not is condemned already, because he hath not believed in the name of the only begotten Son of God. ... He that believeth on the Son hath everlasting life: and he that believeth not the Son shall not see life; but the wrath of God abideth on him.
Rom. 11:16-21 For if the firstfruit be holy, the lump is also holy: and if the root be holy, so are the branches. And if some of the branches be broken off, and thou, being a wild olive tree, wert graffed in among them, and with them partakest of the root and fatness of the olive tree; Boast not against the branches. But if thou boast, thou bearest not the root, but the root thee. Thou wilt say then, The branches were broken off, that I might be graffed in. Well; because of unbelief they were broken off, and thou standest by faith. Be not highminded, but fear: For if God spared not the natural branches, take heed lest he also spare not thee.

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Edition & Source Public domain

Source: The Heidelberg Catechism
Translation: English translation as published by apuritansmind.com; Scripture proofs: King James Version
Published: First published 1563
Original URL: apuritansmind.com/creeds-and-confessions/the-heidelberg-catechism-by-zacharias-ursinus/
Copyright: Public domain.
License: Public domain
Attribution: Public domain. Text transcribed from apuritansmind.com, which reformed.org's historic-documents index links to. Compared automatically with the PDF at heidelberg-catechism.com, which is the same translation family: 217 of 258 questions and answers matched. Of the 41 that did not, I checked 18 by hand: Q12, Q18, Q27, Q56 and Q61 differ in wording, and the others were page-layout noise in the PDF; the remaining 23 were not checked by hand. Not yet proofread against a printed edition. Part divisions are the standard ones (Q1–2 introduction; Q3–11 misery; Q12–85 deliverance; Q86–129 thankfulness). The question 23 creed text had verse-number markers removed. Scripture proof texts are the King James Version, from the public-domain Project Gutenberg edition, added with tools/add_kjv.py. A few references could not be matched to verses (whole-chapter references such as ‘Genesis 1 throughout’, and typing slips in the source) and are shown without text.

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Cite as The Heidelberg Catechism. First published 1563. https://www.apuritansmind.com/creeds-and-confessions/the-heidelberg-catechism-by-zacharias-ursinus/