Is MySQL logic evaluation lazy/short-circuiting in JOIN clause?

Take the following expression: FALSE AND (expression)

Will MySQL evaluate the expression or just move on as soon as it sees FALSE?

Some background context-- I wanted to speed up a query by doing:

JOIN... ON (indexed_column1=indexed_column2 AND non_indexed_column_a=non_indexed_column_b)

For background on why I'm doing this query see this answer

If it's going to always evaluate non_indexed_column_a=non_indexed_column_b then no time is saved with that.