Base Characterization of Tychonoff Spaces

Question

$X$ is a Tychonoff space iff it has a base $\mathscr B$ such that -

1. For every $x\in X$ and every $U\in\mathscr B$ containing $X$, there exists a $V\in\mathscr B$ such that $x\not\in V$ and $U\cup V = X$
2. For any $U,V\in\mathscr B$ where $U\cup V = X$, there exist $U',V'\in\mathscr B$ such that $V^c\subseteq U'$, $U^c\subseteq V'$, and $U'\cap V' = \varnothing$

Hint: Modify the proof of the Urysohn theorem.

I was able to show that $X$ has to be regular for the reverse implication, but was not able to proceed any further. Any help would be appreciated!