EK at EWR checked in for A3. IDK whether they will do the same for LY.
LY at TLV checked in for EK. This might indicate that EK could do the same.
I've done the ATH connection from and to LY many times in the past few months
If the LY portion is on the same ticket you're lucky
From TLV my bags always connected to the next airline, no issues
However to TLV, you should tell LY security in ATH that you have a connecting bag and show them the tag. The sooner the better to allow them to process it and allow it on the flight.
If the LY portion is a separate ticket, that may depend on the competency of the check in agent for you first flight. Sometimes its even best to walk away and try with a different agent. They can manually add the LY flight, but not always....
EK can, but sometimes don't.... (I had to try 3 agents in HKG 2 weeks ago)