Interpreting predicate formulas in the structure of arithmetic

Question

Given two formulas

a) $(\forall x)(\phi(x)\rightarrow\varphi)\;\;\;\;\;\;$

b)$(\forall x)\phi(x)\rightarrow\varphi\;\;\;\;\;$

Let $\;\mathbb{S}=(\mathbb{N},+,\times,\le,0,S)$ (where $S$ stands for successor e.g $S(0)=1$) be the language of arithmetic.

Im told that if we let $\phi(x):x=2$ and $\varphi:2+2=5\;$then (a) is false wheras (b) is true, obviously $\varphi$ is false in the structure $\mathbb{S}$, but interpreting these formulas in this way just makes no sense to me! Anybody care to explain why (a) is true and (b) is false?

Answer 1

It's about the parentheses.

When substituting the given formulas, a) reads as

For all $x$: $\ x=2$ implies $2+2=5$.

This is false.
Whereas b) reads as

If (for all $x$ we have $x=2$), then $2+2=5$.

This is true as the condition is false.