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In propositional Gödel logic, one can imagine a new operator $=$ as

$V(\phi = \psi) = \begin{cases}1,&\text{iff }V(\phi) =V(\psi)\\0,&\text{otherwise}\end{cases}$

with $V$ being the corresponding interpretation.

My question is if someone could imagine a formulas made up from the classic connectives($\land$,$\to$, $\lor$,$\neg$) which would be equivalent for this.

My first consideration was simply the connective $\leftrightarrow$ which fulfills this property in classical logic. But in case of Gödel(fuzzy) logic, it doesn't hold for the definition for 0.

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Over Gödel logic, your operator is interdefinable with Baaz’s delta: $\phi=\psi$ can be defined as $\Delta(\phi\leftrightarrow\psi)$, and conversely, $\Delta\phi$ can be expressed as $\phi=\top$. It is not definable in Gödel logic alone.

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    Additionally, would it be possible to define $\phi > \psi$ with the respective meaning as well? – 2017-02-01
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    Well, $\phi\le\psi$ can be expressed as $\Delta(\phi\to\psi)$, and then $\phi>\psi$ can be written as $\neg(\phi\le\psi)$. Again, Baaz delta is unavoidable here, as $\Delta\phi$ can be conversely expressed as $\neg(\top>\phi)$. – 2017-02-01