Uniform continuity and the order of quantifiers

Question

I’m taking my first course in real analysis, and I’m trying to prove the following proposition.

Proposition: If $f:S\to\mathbb{R}$ is uniformly continuous, then $f$ is continuous.

In comparing seemingly similar definitions of continuity and uniform continuity, I noticed that the only difference in the definition is the position of the universal quantification $\forall c\in S$.

1. $f$ is continuous if $\forall c\in S, \forall \epsilon>0, \exists \delta>0$ such that $\forall x\in S$, if $|x-c|<\delta$ then $|f(x)-f(c)|<\epsilon$.
2. $f$ is uniformly continuous if $\forall \epsilon>0, \exists \delta>0 \text{ such that } \forall x,c\in S$, if $|x-c|<\delta$ then $|f(x)-f(c)|<\epsilon$.

Leaving only logically relevant symbols,

1. Continuity: $\forall c \forall \epsilon \exists \delta \forall x P(x, c, \epsilon, \delta)$
2. Uniform continuity: $\forall \epsilon\exists\delta\forall x \forall c P(x, c, \epsilon,\delta)$

where $P(x,c,\epsilon,\delta)$ is “if $|x-c|<\delta$, then $|f(x)-f(c)|<\epsilon$”

Now, my question is whether it is valid to say that in general:

if $\exists x \forall y P(x,y)$ is true, then so is $\forall y \exists xP(x,y)$.

If this is the case, then I think I can claim that uniform continuity implies continuity, because what I did is essentially to move the $\forall c$ quantification from the right of $\exists\delta$ to its left.