Let $M$ be a von Neumann algebra and $a,b\in M$. Suppose $a\geq b$ and $a$ is equivalent to $c\in M$.
Then there exists $d\in M$ such that $d$ is equivalent to $b$ and $c\geq d$.
Since $a$ is equivalent to $c$, there exists a partial isometry $u\in M$ such that $a=u^*u,c=uu^*$. So we have $uau^*\geq ubu^*$, and thus $c=uu^*\geq ubu^*$. If we let $d：=ubu^*$, how to check that $d$ is equivalent to $b$? If the above proof is not correct, how to find $d$ such that $d$ is equivalent to $b$ and $c\geq d$.
What you don't mention is that $a,b,c,d$ are projections.
You have $d=ubu^*=(bu^*)^*bu^*$. So $d$ is equivalent to $$ bu^*ub=bab=b^2=b. $$