Heyting-algebra meaning

A bounded lattice equipped with a binary operation a→b of implication such that (a→b)∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b.

If the less-than-or-equal relation in the definition of Heyting algebra is interpreted to mean logical entailment, then (a→b)∧a ≤ b translates to "(a→b), a ⊢ b" which is Modus Ponens. The formulae in the second condition translate to "c, a ⊢ b ⇒ c ⊢ a→b" which is the Deduction Theorem.