God sometimes uses the words "Israel," "Judah," and "Jerusalem" when referring to the church. We have to examine the context of the passage to determine which church is in view: the earthly organization or the eternal church made up of only true believers.
Israel = Jerusalem = Judah
First, we must recognize that God often uses the terms "Israel," "Jerusalem," and "Judah" interchangeably.
God refers to the vineyard as the house of Israel:
"For the vineyard of the LORD of hosts is the house of Israel, and the men of Judah his pleasant plant: and he looked for judgment, but behold oppression; for righteousness, but behold a cry." (Isaiah 5:7)
But He also refers to it as the inhabitants of Jerusalem:
"Therefore thus saith the Lord GOD; As the vine tree among the trees of the forest, which I have given to the fire for fuel, so will I give the inhabitants of Jerusalem." (Ezekiel 15:6)
God indicated that He would cause the rain to stop on the vineyard, which He refers to as the house of Israel in the following passage:
"And I will lay it waste: it shall not be pruned, nor digged; but there shall come up briers and thorns: I will also command the clouds that they rain no rain upon it. For the vineyard of the LORD of hosts is the house of Israel, and the men of Judah his pleasant plant: and he looked for judgment, but behold oppression; for righteousness, but behold a cry." (Isaiah 5:6-7)
Elsewhere, we find this prophecy coming to pass in Judah:
"Judah mourneth, and the gates thereof languish; they are black unto the ground; and the cry of Jerusalem is gone up. And their nobles have sent their little ones to the waters: they came to the pits, and found no water; they returned with their vessels empty; they were ashamed and confounded, and covered their heads. Because the ground is chapt, for there was no rain in the earth, the plowmen were ashamed, they covered their heads." (Jeremiah 14:2-4)
Notice in the above passage we also find God using "Judah" and "Jerusalem" interchangeably.
God indicated that He would break down the wall of the vineyard and that the vineyard would be trodden down, which above we found to be the house of Israel:
"And now go to; I will tell you what I will do to my vineyard: I will take away the hedge thereof, and it shall be eaten up; and break down the wall thereof, and it shall be trodden down" (Isaiah 5:5)
Elsewhere, we learn that the walls of Jerusalem were broken down:
"And they burnt the house of God, and brake down the wall of Jerusalem, and burnt all the palaces thereof with fire, and destroyed all the goodly vessels thereof." (2 Chronicles 36:19)
We also read about Jerusalem being trodden down:
"And they shall fall by the edge of the sword, and shall be led away captive into all nations: and Jerusalem shall be trodden down of the Gentiles, until the times of the Gentiles be fulfilled." (Luke 21:24)
Israel Signifies the Church
The nation of Israel was a picture of the true believers. God rescued them from captivity in Egypt and brought them safely into the Promised Land. This foreshadowed the liberty that is brought to the true believers who God rescues from the bondage of sin as He brings them safely into His Kingdom:
"Stand fast therefore in the liberty wherewith Christ hath made us free, and be not entangled again with the yoke of bondage." (Galatians 5:1)
We read that Israel will be redeemed from all its sins:
"And he shall redeem Israel from all his iniquities." (Psalm 130:8)
We know that most of those in national Israel were not saved and therefore not redeemed. The only group we could say are entirely redeemed from all their sins is the true believers. We read that all of Israel will be saved:
"And so all Israel shall be saved: as it is written, There shall come out of Sion the Deliverer, and shall turn away ungodliness from Jacob" (Romans 11:26)
But we know that not every person in national Israel was saved. This is a reference to the true believers.
God distinguishes between two different Israels:
"Not as though the word of God hath taken none effect. For they are not all Israel, which are of Israel: Neither, because they are the seed of Abraham, are they all children: but, In Isaac shall thy seed be called. That is, They which are the children of the flesh, these are not the children of God: but the children of the promise are counted for the seed." (Romans 9:6-8)
God refers to this spiritual Israel as the Israel of God:
"And as many as walk according to this rule, peace be on them, and mercy, and upon the Israel of God." (Galatians 6:16)
As citizens of the Israel of God, believers are spiritually referred to as Jews:
"For he is not a Jew, which is one outwardly; neither is that circumcision, which is outward in the flesh: But he is a Jew, which is one inwardly; and circumcision is that of the heart, in the spirit, and not in the letter; whose praise is not of men, but of God." (Romans 2:28-29)
And of course the Israel of God has Christ as its King and Governor:
"Let Christ the King of Israel descend now from the cross, that we may see and believe. And they that were crucified with him reviled him." (Mark 15:32)
"And thou Bethlehem, in the land of Juda, art not the least among the princes of Juda: for out of thee shall come a Governor, that shall rule my people Israel." (Matthew 2:6)
Christ came to seek the lost sheep of Israel:
"But he answered and said, I am not sent but unto the lost sheep of the house of Israel." (Matthew 15:24)
The sheep are those who belong to Christ; they are the believers:
"My sheep hear my voice, and I know them, and they follow me" (John 10:27)
Therefore, in the phrase "sheep of Israel" the Israel in view is spiritual Israel.
God also uses the label "Jerusalem" when referring to the spiritual kingdom that is made up of all of the true believers:
"But Jerusalem which is above is free, which is the mother of us all." (Galatians 4:26)
"But ye are come unto mount Sion, and unto the city of the living God, the heavenly Jerusalem, and to an innumerable company of angels" (Hebrews 12:22)
"Him that overcometh will I make a pillar in the temple of my God, and he shall go no more out: and I will write upon him the name of my God, and the name of the city of my God, which is new Jerusalem, which cometh down out of heaven from my God: and I will write upon him my new name." (Revelation 3:12)
The true believers make up the eternal church of God. However, sometimes God uses the terms "Israel," "Jerusalem," and "Judah" to refer to the New Testament church:
"Behold, the days come, saith the LORD, that I will make a new covenant with the house of Israel, and with the house of Judah" (Jeremiah 31:31)
The nation of Israel ceased to be the earthly representation of God's people at the cross. This role was transferred to the New Testament church referred to here as the house of Israel. Numerous other examples of this can be found when passages are studied in detail.