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

Theorem
Let $\struct {S, \vee, \wedge, \preceq}$ 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 $d: S \to S$:
 * $\forall s \in S: \map d s = a \wedge s$

Define a mapping $g: S \to S$:
 * $\forall s \in S: \map g s = a \to s$

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

By definition of Galois connection:
 * $x \preceq \map g y$ $\map d x \preceq y$

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