On a related note, John Piper, in Reason #30 Why Jesus Came to Die:
But there is another sense in which we die precisely because Christ died in our place and for our sins. “He himself bore our sins in his body on the tree, that we might die . . .” (1 Peter 2:24). He died that we might live; and he died that we might die. When Christ died, I, as a believer in Christ, died with him. The Bible is clear: “We have been united with him in a death like his” (Romans 6:5). “One has died for all, therefore all have died” (2 Corinthians 5:14).