The context tells us what ''equality with God'' means. In John 5:18, the Jews wanted to stone Jesus because they believed Jesus broke the sabbath just because he healed the sick man. Verse 16 said ''the Jews were persecuting Jesus, because he was doing these things (i.e. healing the sick) on the Sabbath''. But this wasn't the only reason why the Jews wanted to stone Jesus. The other reason actually made the Jews to seek ''all the more to kill him'' and that's ''calling God his own father'' meaning ''making himself equal with God''.
When Jesus calls God his ''own Father'', Jesus is claiming to be God's ''own son''. The corollary of God being Jesus’ ''own father'' is that Jesus is God’s ‘own son’’. One of the earliest N.T. text indeed spoke of Jesus as God’s ‘’own son’’ (Grk. idiou huiou Romans 8:32 ). Jesus understood himself to be the ''only son of his kind'' (Grk. monogenes huios) (John 3:16, 18).
In just one verse earlier (v. 17), Jesus did call God ''my father'' (Grk. ho pater mou) and said that his father was ''working until now'' (Grk. heos arti egazetai) and he (Jesus) himself was also working: ''and I am working '' (Grk. kargo ergazomai). Jesus was claiming to be equal with God in doing good works (in context, the works were specifically referring to the healing of the sick man on sabbath).
In Luke 6:34, the Greek word isos refer to ''the same thing'' (i.e. referring to the same ''amount'' in context). in John 5:18, the Greek word isos refer to ''the same thing'' (i.e. the same works) which the Father and the Son do.
Jesus was not doing it alone. Jesus was ''[doing] the same thing [i.e. breaking the Sabbath, by healing the sick man] with God''. The Jews couldn't accept the words of Jesus because according to Jesus, he was not alone doing the works but that God himself as working (i.e. by healing the sick man, which was for the Jews, it meant breaking the sabbath) (v.17) and that he (Jesus) cannot do the works (i.e. healing the sick man/ break the sabbath) unless he sees the Father himself does it (v.19). The Jews saw it as Jesus speaking evil of God (i.e. blasphemy) that is why they sought all the more to kill Jesus.
Based on verse 19, Jesus claimed that he was able to do ‘’nothing’’ (Greek: ouden) about the Sabbath. He wasn’t changing the Sabbath, adding his own ideas to it. Jesus was merely obeying the sabbath ‘’in the same way’’ (Greek: homoia) or in the original way it was given by God.
''So Jesus said to them, “Truly, truly, I say to you, the Son can do nothing of his own accord, but only what he sees the Father doing. For whatever the Father does, that the Son does likewise.'' (John 5:19 ESV)
Jesus believed that the Father himself would have broken the sabbath [of the current Jewish understanding]. Jesus was explaining that what he had done (i.e. healing the sick on a sabbath day) was something God himself would do if God were in such a scenario. This is why Jesus claimed that he could not have broken it (i.e. i myself can do nothing). Breaking the sabbath wasn’t something Jesus could have done without a basis. Jesus explained that he could not have done it if he didn’t see God himself doing it. God is doing good on sabbath days. This is what God showed Jesus and this is what Jesus will do. And this is what Jesus had done on the sabbath day.
Jesus did break the sabbath, the sabbath which the Jews themselves knew, not the sabbath which God himself knew. God knows exactly what the sabbath truly means. The sabbath wasn’t what the Jews think that it really was. Jesus was doing good on a sabbath day. That’s not breaking the sabbath but fulfilling it. For Jesus, the sabbath is not only a rest day but also a day to do good deeds.
John 5:1-19 (ESV):
5 After this there was a feast of the Jews, and Jesus went up to Jerusalem. 2 Now there is in Jerusalem by the Sheep Gate a pool, in Aramaic[a] called Bethesda,[b] which has five roofed colonnades. 3 In these lay a multitude of invalids—blind, lame, and paralyzed.[c] 5 One man was there who had been an invalid for thirty-eight years. 6 When Jesus saw him lying there and knew that he had already been there a long time, he said to him, “Do you want to be healed?” 7 The sick man answered him, “Sir, I have no one to put me into the pool when the water is stirred up, and while I am going another steps down before me.” 8 Jesus said to him, “Get up, take up your bed, and walk.” 9 And at once the man was healed, and he took up his bed and walked.Now that day was the Sabbath. 10 So the Jews[d] said to the man who had been healed, “It is the Sabbath, and it is not lawful for you to take up your bed.” 11 But he answered them, “The man who healed me, that man said to me, ‘Take up your bed, and walk.’” 12 They asked him, “Who is the man who said to you, ‘Take up your bed and walk’?” 13 Now the man who had been healed did not know who it was, for Jesus had withdrawn, as there was a crowd in the place. 14 Afterward Jesus found him in the temple and said to him, “See, you are well! Sin no more, that nothing worse may happen to you.” 15 The man went away and told the Jews that it was Jesus who had healed him. 16 And this was why the Jews were persecuting Jesus, because he was doing these things on the Sabbath. 17 But Jesus answered them, “My Father is working until now, and I am working.” 18 This was why the Jews were seeking all the more to kill him, because not only was he breaking the Sabbath, but he was even calling God his own Father, making himself equal with God. 19 So Jesus said to them, “Truly, truly, I say to you, the Son can do nothing of his own accord, but only what he sees the Father doing. For whatever the Father[e] does, that the Son does likewise. 20 For the Father loves the Son and shows him all that he himself is doing. And greater works than these will he show him, so that you may marvel.
Conclusion
According to the context (John 5:1-17, 5:19), Jesus was ''equal with God [in doing good works on Sabbath] in John 5:18.
