The departure of the two barrister-bishops comes four weeks after the report into child sexual abuse in the Dublin archdiocese found that it had shielded clergy who criminally abused children from the law.
In a joint statement, released before midnight on Christmas Eve, the bishops said: "We, Bishop Eamonn Walsh and Bishop Raymond Field, have this evening informed Archbishop Diarmuid Martin that we are offering our resignation.
"As we celebrate the feast of Christmas, the birth of our saviour, the prince of peace, it is our hope that our action may help to bring the peace and reconciliation of Jesus Christ to the victims/survivors of child sexual abuse. We again apologise to them."
The statement added: "Our thoughts and prayers are with those who have so bravely spoken out and those who continue to suffer in silence."
Bishop Drennan is now the sole prelate named in the damning Murphy report who has yet to offer his resignation.
As he arrived at Galway Cathedral to celebrate Mass yesterday, Bishop Drennan said: "I have nothing to say at the moment. I'm concentrating on preparing Mass."
But the bishop, who last week publicly insisted he would not be standing down, is expected to confirm his own resignation amid pressure from Dr Martin.
Dr Martin last night said he respected the decision of Bishops Walsh and Field.
He said the two prelates were "extremely good bishops" and had done many things in the Dublin archdiocese for which people were grateful. But he said good people had to be accountable.
He refused to get involved in any discussion about the future of Bishop Drennan.
The latest resignations came after the Irish Independent exclusively reported on Thursday that Dr Martin faced down a behind-the-scenes attempt by Bishop Walsh to rally the support of the clergy.
In a confidential letter, written as part of a lobby campaign among Dublin priests, Bishop Walsh -- who was previously tipped as a future archbishop -- repeatedly said he did no wrong and that it would be an injustice for him to step down. But his position became untenable after Dr Martin refused to back his senior assistant.
On Wednesday, the Bishop of Kildare and Leighlin, Jim Moriarty, went a step further after he accepted the principle of collective responsibility for failing to persuade his former boss, Cardinal Desmond Connell, to investigate the criminal history of notorious paedophile priest 'Fr Edmondus' in a secret diocesan file.
Abuse victims yesterday welcomed the latest resignations.
He added: "It is important for me to live in a society where those who have subordinated the welfare of vulnerable children to the betterment of any institution learn that such behaviour is not acceptable."
The Murphy inquiry, based on a sample 46 priests, revealed a catalogue of paedophilia and subsequent cover-ups over three decades -- because the Catholic hierarchy was obsessed with secrecy and granted immunity by gardai.
In his homily at the Christmas Eve Vigil Mass in St Mary's Pro-Cathedral in Dublin -- shortly before the resignations of his two auxiliary bishops -- Dr Martin said it had been "a painful year" for the Church.
"But the church today may well be a better and safer place than . . . 25 years ago," he added.
"The Church in Dublin is called to conversion and to renewal."
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