First, Williams says: “If a writer teaches that Christ effectively dealt with the penalty of sin deserved from God when he died bearing it, then he is teaching that his death made satisfaction.”
So that's a preliminary definition of PSA. That Christ not only paid for our sins but satisfied God's wrath for all the elect for all time.
Now, he specifies: "An author can be held to teach the Penal doctrine if he plainly states that the punishment deserved by sin from God was borne and dealt with by Jesus Christ in his death on the cross.”
He first quotes Justin Martyr. He wrote to this man named Trypho who wondered how Christ could have been cursed on the tree. Somehow, Flood thought that Justin agreed that Christ wasn't cursed for our sins, but what Justin writes is contrary.
"When Justin says that no curse was on the Christ, the Messianic reference is pointed. He is saying that the cross is no obstacle to identifying Jesus as the Christ because he did not of himself deserve the curse. He is not making an absolute statement denying that the Christ was in any sense cursed, since in this very section he states plainly that he took the legal curses of ‘the whole human family’ upon himself." In fact, Justin cites the verse in Galatians and Deuteronomy where it says, "Cursed is any man hung on a tree." Though not deserving them, He took all our curses and suffered God's wrath so we would not have to.
In referring to Eusebius, Williams claims that Flood believes him to have considered Christ's death as a sympathy, not as a complete substitution. In fact, it is speculated that Flood believes that, but Williams sees no where that Eusebius believed that. "I cannot think of a single advocate of penal substitutionary atonement who denies the idea of Christ conquering death.”
Then he refers to Eusebius's writings: "In the introduction to the book, Eusebius states that Christ as God offered sacrifice ‘propitiating the Father, that as man he sacrificed the firstfruits of the human race like a lamb ‘to the Father, and that it was necessary that this lamb of God ‘should be offered as a sacrifice to God."
Then, Williams moves on to my all-time favorite: Athanasius. Quoting the man himself, "
“He took to himself a body which could die, in order that, since this participated in the Word who is above all, it might suffice for death on behalf of all, and because of the Word who was dwelling in it, it might remain incorruptible, and so corruption might cease from all men by the grace of the resurrection.” See, Athanasius believed that Christ's death satisfied God's wrath for the elect for all time.
He cites Gregory of Nanzianzus who also explains Christ's death as not merely erasing God's anger at humanity, but satisfying it.
And finally, my friend Augustine. He allegorizes the snake lifted up in the desert by Moses that took away the Israeli snake bites because Jesus referred to it when talking to Nicodemus. He shows that just as the diseases were transferred to the bronze snake, so were our sins and infirmities transferred to Jesus. "It is not removed by the abolition of the entire penal system but is transferred to Christ (note the repeated use of transfero) and actually transpires in him (ut esset in). It is removed, but only by being fulfilled."
So in conclusion, penal substitutionary atonement teaches that Jesus's crucifixion satisfied God's wrath against his elect for all time. None of those people will be in hell because that would be a double jeopardy. And Jesus did not just erase God's penal system, he fulfilled it. And that is how he saved his Bride.