A common way to refer to the unity of believers was to call them the body of Christ. Who first wrote the phrase?

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Paul in the Epistle to the Corinthians first used the phrase. He used it often in his other Epistles.

Source: 1^st Corinthians 10:16

How does unity affect our prayers to God?

According to Jesus it helps them when we agree God hears.

Source: Matthew 18:19

While all of the apostles agreed with unity not all mentioned it. Who was the other Apostle to mention unity in his epistles?

The other apostle who called for unity is Peter.

Source: 1^st Peter 3:8

Paul often appealed for unity in the church. What was the first letter to call for that?

The first letter the the church at Corinth called for unity. He opposed division in the church.

Source: 1^st Corinthians 1:10-13

Where did Jesus pray for the unity of Christians?

He prayed for the unity of all in the garden of Gethsemene.

Source: John 17:20-21