But there's just one problem with that phrase: it's not true. As Christians we know that our birth into eternal life is not grounded in our death, it's grounded in Jesus' death. Eternal life doesn't begin when life in this world ends. It begins when we are crucified with Christ in baptism and raised with Him to a new life by water and the Word. (Galatians 2:20; John 3:5; Titus 3:5)
We have eternal life already right now through faith in Jesus. "Whoever believes in the Son has eternal life" (John 3:36). That's not just a future but a present reality. The eternal life we have in Christ begins in this life, continues through our earthly death, and is consummated on the Last Day when our bodies are raised with Christ to share in His glory (Philippians 3:21; 1 John 3:2). This is how Jesus can say that those who live and believe in Him will never die (John 11:26). It is true that those who have died in the Christian faith are no longer burdened by the troubles of this world and our fallen nature. But they already were born to eternal life, ever since they believed and were baptized.
If we're going to talk about any future (re)birth, let us have in mind the resurrection of the body at Christ's return. And if we're going to talk about being born to eternal life when we die, let the reference of that death be your baptismal burial and resurrection with Jesus. "We were buried therefore with Him by baptism into death, in order that, just as Christ was raised from the dead by the glory of the Father, we too might walk in newness of life. For if we have been united with Him in His death, we shall certainly also be united with Him in His resurrection" (Romans 6:4-5).