What did the early church fathers (pre-5th c.) teach with regards to the doctrine of baptism as a necessity for salvation? The title says it all, really. 

What did the ECF believe and teach with regards to baptism being or not being necessary for salvation?

Related: What did the early church fathers (pre-5th c.) teach with regards to baptismal regeneration?
“First of all it bids us bear in mind that we have received baptism
  for the remission of sins, in the name of God the Father, and in the
  name of Jesus Christ, the Son of God, who was incarnate and died and
  rose again, and in the Holy Spirit of God. And that this baptism is
  the seal of eternal life, and is the new birth unto God, that we
  should no longer be the sons of mortal men, but of the eternal and
  perpetual God.”


-Irenaeus, Demonstration of the Apostolic Preaching Ch. 3 


  “Those, then, who, after having been baptized into Father, Son, and
  Holy Spirit, and having been taught that there is one divine nature in
  three subsistences, are rebaptized, these, as the divine Apostle says,
  crucify the Christ afresh (chapter 3)… But those who were not baptized
  into the Holy Trinity, these must be baptized again.” (chapter 9)


-John of Damascus, An Exposition of the Orthodox Faith, Book IV


  “It is possible that a man is wishing to pay court to a woman, and
  came hither on that account. The remark applies in like manner to
  women also in their turn. A slave also perhaps wishes to please his
  master, and a friend his friend. I accept this bait for the hook, and
  welcome thee, though thou camest with an evil purpose, yet as one to
  be saved by a good hope. Perhaps thou knewest not whither thou wert
  coming, nor in what kind of net thou art taken. Thou art come within
  the Church's nets: be taken alive, flee not: for Jesus is angling for
  thee, not in order to kill, but by killing to make alive: for thou
  must die and rise again.” (5)


-Cyril of Jerusalem’s Catechetical Lectures, Lect. 1


  “Great is the Baptism that lies before you: a ransom to captives; a
  remission of offences; a death of sin; a new-birth of the soul; a
  garment of light; a holy indissoluble seal; a chariot to heaven; the
  delight of Paradise; a welcome into the kingdom; the gift of
  adoption!” (16)


-Cyril of Jerusalem’s Catechetical Lectures, Lect. 1


  “We may not receive Baptism twice or thrice; else it might be said,
  Though I have failed once, I shall set it right a second time: whereas
  if thou fail once, the thing cannot be set right…” (7)


-Cyril of Jerusalem’s Catechetical Lectures, Lect. 1


  “Jesus sanctified Baptism by being Himself baptized. If the Son of God
  was baptized, what godly man is he that despised Baptism? But He was
  baptized not that He might receive remission of sins, for He was
  sinless; but being sinless, He was baptized, that He might give to
  them that are baptized a divine and excellent grace. For since the
  children are partakers of flesh and blood, He also Himself likewise
  partook of the same, that having been made partakers of His presence
  in the flesh we might be made partakers also of His Divine grace: thus
  Jesus was baptized, that thereby we again by our participation might
  receive both salvation and honor (11)… For thou goest down into the
  water, bearing thy sins, but the invocation of grace, having sealed
  thy soul, suffereth thee not afterwards to be swallowed up by the
  terrible dragon. Having gone down dead in sins, thou comest up
  quickened in righteousness. For if thou hast been united with the
  likeness of the Saviour's death, thou shall also be deemed worthy of
  His Resurrection. For as Jesus took upon Him the sins of the world,
  and died, that by putting sin to death He might rise again in
  righteousness; so thou by going down into the water, and being in a
  manner buried in the waters, as He was in the rock art raised again
  walking in newness of life (12)… Moreover, when thou hast been deemed
  worthy of the grace, He then giveth thee strength to wrestle against
  the adverse powers. For as after His Baptism He was tempted forty
  days…” (13)


-Cyril of Jerusalem’s Catechetical Lectures, Lect. 3