In Paul Cohen's
*Set Theory and the Continuum Hypothesis*
(Page 71) there is a lemma with the assumption that
$\exists x\, P(x)$.
The ''proof'' there, uses the following argument:

Intuitively the lemma is obvious, since if $F$ is valid it holds for every set $S$ in which relations and constants corresponding to the formal language are defined and hence for the subset $$\{x|\,x \in S \land P(x)\}.$$ Thus $F_P$ is true in every model and hence is valid.

Where

If $F$ is a formula, then $F_P$ denotes the formula obtained from $F$ by adjoining to every variable $x$ the condition $P(x)$. That is in building $F_P$, each $\exists x\, B$ becomes $$\exists x\,[P(x) \land B]$$ and each $\forall x\, B$ becomes

$$\forall x\,[P(x) \rightarrow B].$$ We say $F_P$ is $F$relativizedto the condition $P(x)$.

Obviously, I must be missing something. But say that $F$ is the following statement $$\exists x\, \lnot P(x)$$ and it holds in some model $M$. That is $P(x)$ is true for some $x\in M$ and is false for some $x\in M$. Now for the subset $$B = \{x|\,x \in M \land P(x)\}$$ $F$ clearly fails and $F_P$ cannot be true.

What am I missing? Must $F$ be restricted to certain types of statements?

Christian Remling and Noah S. Explained that $F$ as defined above is not valid.

Still, I would like to get a better intuition of the lemma mentioned in Paul Cohen's book. So let's look at another example.

Assuming Peano axioms, say $F$ is $$\forall x \exists y\, x < y $$ is it considered valid?

If $F$ is valid, then if $P(x)$ is some property like $$x < 5$$ then $F_p$ becomes $$\forall x\, [x < 5 \rightarrow \exists y\, (y < 5 \land x < y)].$$ Which is clearly invalid considering $x=4$.

If $F$ is *not* valid, can someone give
a ''non trivial'' example of a valid statement
that includes $\forall$?