# Barendregt's Variable Convention: what does it mean?

Barendregt's Variable Convention: If $M_1,...,M_n$ occur in a certain mathematical context (e.g. definition, proof), then in these terms all bound variables are chosen to be different from the free variables.

This convention appears a lot in papers and I always wonder what does it mean. I want to know exactly what it is saying. I have two ideas about it

when you working on the theory

1. always keep a bound variable's name different from the names of variables at a particular time (Note:two bound variables may have same names)

2. always keep a bound variable name different from name of other bound variables name and all free variable names (every name is distinct)

Variable convention does not mention about if we can have two bound variables with same names, but I feel variable convention also implies such distinctiveness of bound variables.

So anyone clarify it to me?