What's the name of this equality axiom?

Question

For any numbers $a$, $b$, and $c$, the transitive property of equality states that if $a=b$ and $b=c$, then $a=c$. In a slight variation, for any numbers $a$ and $c$, and for some number $b$, then $a=b$ and $b=c$ if and only if $a=c$. Is there a special name for this property?