Does $\forall x((\forall zR(x,z) \rightarrow \exists yR(y,x)))$ holds in any non-empty model?

Question

I am trying to prove or disprove that the formula $\forall x((\forall zR(x,z) \rightarrow \exists yR(y,x)))$ is true in any non-empty model with a predicate R? (the formula is vacuously true when the model is empty set).
I tried using logical equivalences to simplify the formula:
$\forall x((\forall zR(x,z) \rightarrow \exists yR(y,x))) \equiv \forall x((\exists z\space \neg R(x,z) \vee \exists yR(y,x)) \equiv \forall x\exists z\exists y((\neg R(x,z) \vee R(y,x))$
Then my intuition said that if our model is not empty for all x, we can choose $y=x$ and $z=x$; therefore, the statement holds for any non-empty model, however, I've never encountered such a case before so I don't know if my reasoning is correct here, I'll appreciate any insights, thanks !

Answer 1

As noted in the comment, there is a syntax error (unbalanced parentheses) in the starting formula.

(1) If the syntax error is resolved as $$\forall x(\forall z(R(x,z)\rightarrow\exists yR(y,x)))$$

for any $\alpha$ and $\beta$ in the domain,

$$\neg R(\alpha,\beta)\vee\exists yR(y,\alpha)$$

must be satisfied. The first disjunct tells that no member of the domain is related to another by $R$ (i.e., $R$ is an empty relation) and the second disjunct tells that at least a $y$ exists that is related to any member of the domain.

In case that $R$ is not an empty relation, there must be a member related to any other in the domain, which is not satisfiable in general.

(2) Else, if the syntax error is resolved as the OP carries on $$\forall x(\forall zR(x,z)\rightarrow\exists yR(y,x))$$

we have $$\forall x\exists z\exists y(\neg R(x,z) \vee R(y,x))$$

Therefore, for any $\alpha$ in the domain

$$\exists z\exists y(\neg R(\alpha,z) \vee R(y,\alpha))$$

must be satisfied. Hence, there must be at least a $\beta$ that $\alpha$ is not related to or a $\gamma$ that is related to $\alpha$.

To simplify analysis, assign $R$ to an irreflexive relation, such as <, (reflexive relation is easier) and assign numerical values to the rest of the formula, we see that there must exist $y$ or $z$ such that for any $\alpha$ in the domain

$$\alpha\nless z \vee y < \alpha$$

Clearly, the formula is satisfied in general.

For more complex formulas, I recommend semantic tree method.