When Jesus died for our sins it was to take the punishment we deserved for our sins, right? With Jesus taking the punishment we deserve for our sins we therefore are not under punishment for our sins and thus have eternal life, correct? Am I looking at salvation the right way?
Considering what you asked in your OP, I should have posted these two passages as well
(edit: 2 passages and 1 verse ).
Isaiah 53
5 He was pierced through for our transgressions,
He was crushed for our iniquities;
The chastening for our well-being fell upon Him,
And by His scourging we are healed.
6 All of us like sheep have gone astray,
Each of us has turned to his own way;
But the LORD has caused the iniquity of us all
To fall on Him.
Romans 5
8 God demonstrates His own love toward us, in that while we were yet sinners, Christ died for us.
9 Much more then, having now been justified by His blood, we shall be saved from the wrath of God through Him.
10 For if while we were enemies we were reconciled to God through the death of His Son, much more, having been reconciled, we shall be saved by His life.
~Deut
1 Peter 2
24 He Himself bore our sins in His body on the cross, so that we might die to sin and live to righteousness; for by His wounds you were healed.
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