I have to proof that $\Phi$ is true in a model $M$ if and only if ($\forall x$)$\Phi$ is true in $M$.
I read the chapter in my textbook several times and already found this question Proof that a formula $\theta$ is valid if and only if $\forall x \theta$ is valid
But i did not understand this. I guess what i have to proof is, that $\Phi$ in a model is only true if $\Phi$ is true for all possible values of all variables in $\Phi$
I would be very thankful if someone can show me how to proof this. Thank you!
Edit: A formula $\Phi$ of L(R, F, C) is true in the model $M$ = (D, I) for L(R, F, C) provided, $\Phi^{I,A}$ = t for all assignments A. A formula $\Phi$ is valid if $\Phi$ is true in all models for the language. A set S of formulas is satisfiable in M = (D, I), provided there is some assignment A (called a satisfying assignment) such that $\Phi^{I,A}$ = t for all $\Phi$ $\in$ S. S is satisfiable if it is satisfiable in some model.