If two random variables are equal in the distribution, then the $\sigma$-algebra generated by them coincide

Question

Let $X$ in $(\Omega, \mathfrak{F}_1,P_1)$ and $Y$ in $(\Omega, \mathfrak{F}_2,P_2)$ two real random variables such that $X=Y$ in distribution.

The question is:

The $\sigma$-algebras $\sigma_{X}:=\{ A\in \mathfrak{F}_1: A=X^{-1}(B),\mbox{ for all } B\in\mathfrak{B}(\mathbb{R}) \}$ and $\sigma_{Y}$ generated by $X$ and $Y$ are the same?

I think the answer is not, in the case $P_{1} \neq P_2$, but I need to show a counterexample of this.

And in the case $P_1=P_2$ I don't know. Maybe is truth.