Quantification and logical relations, shorthand notation $\forall/\exists x \in M...$

Question

I know the following shorthand: \begin{align*} \exists x \in M : P(x) & := \exists x ( x \in M \to P(x) ) \\ \forall x \in M : P(x) & := \forall x ( x \in M \to P(x) ). \end{align*} Now for me it is obvious that $$ \forall x \in M : P(x) \equiv \neg \exists x \in M : \neg P(x). $$ But also it holds that $\forall x : P(x) \equiv \neg \exists x : \neg P(x)$. So applying I get \begin{align*} \forall x \in M : P(x) & \equiv \forall x ( x \in M \to P(x) ) \\ & \equiv \neg \exists x ( \neg( x \notin M \lor P(x))) \\ & \equiv \neg \exists x : ( x \in M \land \neg P(x) ) \\ \end{align*} and otherwise $$ \forall x \in M : P(x) \equiv \neg \exists x \in M : \neg P(x) \equiv \neg \exists x ( x \in M \to \neg P(x) ). $$ So concluding $$ \neg \exists x : ( x \in M \land \neg P(x) ) \equiv \neg \exists x ( x \in M \to \neg P(x) ). $$ or $$ \exists x : ( x \in M \land \neg P(x) ) \equiv \exists x ( x \in M \to \neg P(x) ). $$ which obviously is not equivalent. But how to explain?

Answer 1

The correct definitions are schematically $$(\forall F x)G x \leftrightarrow \forall x (F x \to G x),$$ and $$(\exists F x)G x \leftrightarrow \exists x (F x \wedge G x),$$ where the variable $x$ is a free variable in $F$ and $G$.

Note carefully that the definition of universal quantification has a conditional statement in its definition, but the definition of existential quantification has a conjunction in its definition. In your specific cases the definitions would read $$\forall x[x \in M \to P(x)],$$ and $$\exists x[x \in M \wedge P(x)],$$ respectively.