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Let $IL$ be intuitionist logic and $\mathcal{L}$ be $IL$ enriched by some set of formulas $\Delta$.

It is also true that $$ IL \vdash \varphi \hspace{10pt} iff \hspace{10pt} \varphi \text{ is valid in every Heyting algebra} $$

I was wondering if this result could be "adapted" into the more general result

$$ \mathcal{L} \vdash \varphi \hspace{10pt} iff \hspace{10pt} \varphi \text{ is valid in every model of } \mathcal{L} $$

Does anyone know if this is true?

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    This is not a research-level question; I think it would be more on-topic at math.stackexchange (and I’d be happy to answer it there). It could also however do with some clarifications: (A) are you considering just *propositional* intuitionistic logic throughout, or first-order (predicate) logic? (B) by “every model of L”, do you mean models in Heyting algebras? (There are many different notions of models of intuitionistic logic, especially if you are talking about the first-order case.) – 2017-01-13

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