How do we know Jesus is the Jehovah who spoke to Moses at the burning bush?

The burning bush voice declared 'I am that I am' to Moses, and Jesus later said, 'Before Abraham was, I am.' Jehovah also identified Himself to Paul as 'I am Jesus,' confirming He is the same Lord who spoke from the bush.

Les explains that the identity is proven by connecting Exodus with John's Gospel. In Exodus, the voice from the burning bush told Moses to tell Israel that 'I am that I am' had sent him. Later, when the Pharisees confronted Jesus, saying He was not yet fifty and claimed to have seen Abraham, Jesus answered, 'Before Abraham was, I am.' Les teaches that Jesus was telling them He is the same one who spoke from the burning bush, the Jehovah of the Old Testament. Similarly, when Paul asked, 'Who art thou, Lord?' or 'Who art thou, Jehovah?' Jehovah answered, 'I am Jesus,' showing He is the God of Abraham, Isaac, and Jacob.

Exodus 3:14

And God said unto Moses, I AM THAT I AM: and he said, Thus shalt thou say unto the children of Israel, I AM hath sent me unto you.

John 8:58

Jesus said unto them, Verily, verily, I say unto you, Before Abraham was, I am.

How does John 8:58 prove that Jesus is the Jehovah of the Old Testament?

In John 8:58 Jesus says, 'Before Abraham was, I am,' echoing the burning bush where God told Moses, 'I am that I am.' That self-designation identified Jehovah. Jesus was therefore claiming to be the same one who spoke from the bush—the Jehovah of the Old Testament.