If $b:[0,T]\times ℝ→ℝ$ is Borel measurable and $(X_t)_{t\in[0,\:T]}$ is continuous and adapted, then $t\mapsto b(t,X_t)$ is progressively measurable

Question

Let

• $T>0$
• $b:[0,T]\times\mathbb R\to\mathbb R$ be $\mathcal B([0,T])\otimes\mathcal B(\mathbb R)$-measurable and $$|b(t,x)-b(t,y)\le L|x-y|\;\;\;\text{for all }t\in[0,T]\text{ and }x,y\in\mathbb R$$ for some $L\ge 0$
• $(X_t)_{t\in[0,\:T]}$ be a real-valued $\mathcal F$-adapted continuous stochastic process on $(\Omega,\mathcal A,\operatorname P)$

Can we show that $$\varphi_t:=b(t,X_t)\;\;\;\text{for }t\in[0,T]$$ is $\mathcal F$-progressively measurable?

It's clear that $X$ itself is $\mathcal F$-progressively measurable. So, it should be easy to conclude the claim. How can we argue?