If joint probabilities equal mean their distributions are equal?

Question

Let $X$ and $Y$ be i.i.d. $N(0, 1)$, and let $S$ be a random sign (1 or -1, with equal probabilities) independent of $(X, Y)$.

\begin{align*} P((SX,SY)∈B)&=P((X,Y)∈B,S=1)+P((−X,−Y)∈B,S=−1) \\ &= P((X,Y)∈B)P(S=1)+P((−X,−Y)∈B)P(S=−1)\\ &= 0.5*P((X,Y)∈B) + 0.5*P((−X,−Y)∈B)\\ &= P((X,Y)∈B) \end{align*}

How can we say that $(SX, SY)$ is multivariate normal?