I don't understand the equivalence between these two logic formulae

Question

I'm having trouble understanding the correction of an exercise. Could someone please explain how the second formula was deduced from the first one ?

$\exists u \exists v \forall x \neg (\neg v = c \rightarrow u \circledcirc (v \circledast x) = c) $

$\exists u \exists v \forall x \neg (\neg v = c \wedge u \circledcirc (v \circledast x) \neq c) $

c is a constant, $\circledcirc$ and $\circledast$ are binary functions.

Thanks a lot !!!

Answer 1

In general, we have the following equivalence:

Implication

$\neg (\phi \rightarrow \psi) \Leftrightarrow (\phi \land \neg \psi)$

Applied to your formula:

$\neg (\neg v = c \rightarrow u \circledcirc (v \circledast x) = c) \Leftrightarrow (\neg v = c \land u \circledcirc (v \circledast x) \not = c)$

If these formulas are part of a larger formula, the equivalence remains. Thus we also have:

$\exists u \exists v \forall x \neg (\neg v = c \rightarrow u \circledcirc (v \circledast x) = c) \Leftrightarrow \exists u \exists v \forall x (\neg v = c \land u \circledcirc (v \circledast x) \not = c) $

So ... you have an extra $\neg$ in the second formula given to you ... the professor must have forgotten to remove it.