Fix a set of variables $X = \{x, y, \dots\}$ and let $S = \{c, f, R\}$ be a first-order signature where $c$ is a constant symbol, $f$ is unary function symbol and $R$ is a binary predicate. Let $A = (D, I)$ be an $S$-structure defined as follows:
$D = \{1, 2, 3\}$
$c^{I} = 3$
$f^{I}(1) = 2$, $f^{I}(2) = 2$, $f^{I}(3) = 3$
$R^{I} = \{(1, 2), (2, 3), (3, 3)\}$.
Consider the formulas $\phi_{1} = R(f(z), c)$ and $\phi_{2} = \exists x \forall y\left(R(f(y), x)\right)$ and the variable assignment $b: X \rightarrow D$ given by $b(x) = 1$ for all $x \in X$.
I'm trying to find out whether $A, b \vDash \phi_{1}$ and whether $A, b \vDash \phi_{2}$. I was able to show that $A, b \vDash \phi_{1}$ but I'm not entirely sure how to find out if $A, b \vDash \phi_{2}$.
I have made an attempt at doing so by showing $A, b \nvDash R(f(y), x)$ and thought this would suffice to show that $A, b \nvDash \phi_{2}$. Is this correct? Thanks in advance!