Inequality with Meet Operation is Equivalent to Inequality with Relative Pseudocomplement in Brouwerian Lattice

Theorem
Let $\left({S, \vee, \wedge, \preceq}\right)$ be s Brouwerian lattice.

Let $a, x, y \in S$.

Then
 * $a \wedge x \preceq y$ $x \preceq a \to y$

Proof
Define a mapping $d: S \to S$:
 * $\forall s \in S: d\left({s}\right) = a \wedge s$

Define a mapping $g: S \to S$:
 * $\forall s \in S: g\left({s}\right) = a \to s$

By Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice:
 * $\left({g, d}\right)$ is Galois connection.

By definition of Galois connection:
 * $x \preceq g\left({y}\right)$ $d\left({x}\right) \preceq y$

Thus by definitions of $g$ and $d$:
 * $a \wedge x \preceq y$ $x \preceq a \to y$