Prenex Normal Form of a Simple Proposition Reads Strangely.

I was trying to convert a simple proposition concerning the real numbers to Prenex Normal Form but arrived at a logical statement that didn't appear equivalent to what I started with.

The proposition I began with:

If two real numbers $a,b$ can get arbitrarily close to one another, then they are equal.

As a logical formula:

if $|a-b|<ε$ for all $ε>0$, then $a=b$.

Entirely with logical symbols:

$ (∀ε>0, (|a-b|<ε)) ⇒ (a=b) $

This logical statement appears identical in meaning to the original proposition above.

Now I perform the following steps to convert this to Prenex Normal Form (PNF) by applying the instructions in https://www.csd.uwo.ca/~lkari/prenex.pdf.

1. Eliminate occurrences of conditionals and biconditionals from the formula. This means removing the $⇒$ symbol in this situation.

   Applying the fact $(p⇒q) ⇔ (¬p∨q)$ gives:

   $ ¬(∀ε>0, (|a-b|<ε)) ∨ (a=b) $

   Observe that the conditional is removed.

2. Move all negations inwards so that negations only appear as part of literals.

   There's only one negation to move and moving it inwards gives:

   $ (∃ε>0, ¬(|a-b|<ε)) ∨ (a=b) $

3. The PNF can now be obtained by moving all quantifiers to the front.

   So we just have to shift the existential quantifier outside and we're done:

   $ ∃ε>0, ( ¬(|a-b|<ε) ∨ (a=b) ) $

   Observe that the formula that comes after the existential quantifier can be rewritten as a conditional since it's in the "$(¬p∨q)$" form. Rewriting the formula as a conditional gives:

   $ ∃ε>0, ( (|a-b|<ε) ⇒ (a=b) ) $

This result does not appear equivalent to what I began with:

If two real numbers 𝑎,𝑏 can get arbitrarily close to one another, then they are equal.

Related phrases:drinker's paradox, drinker paradox, drinker's theorem, Smullyan's Drinker's principle, Raymond Smullyan's "What Is the Name of this Book?"