Related post : What's the difference between material implication and logical implication?
An " implies" sentence is a material conditional sentence that is logically true.
One can use $X \implies Y$ to express the fact that , not only it is not (factually) the case that X is true and Y false, but also that it cannot ( logically) be the case.
For example " $x$ lives in Florida " materially implies " $x$ lives in the USA" for all $x$ ( indeed, factually, it is the case for no $x$ that $x$ lives in Florida while $x$ does not live in the USA). But the former does not logically imply the second, for it could be the case one lives in Florida while not living in the USA (in case Florida had not been joinded to the Union, which is a logically possible situation).
So one cannot write : $ Florida(x) \implies USA(x) $ , you cannot deduce the consequent from the antecedent , using logic alone
- First there is a material conditional formula that is not a sentence
$x^2 = 4 \rightarrow (x=2 \lor x= -2)$.
- Second, you turn it into a sentence using a quantifier
$\forall(x) [x^2 = 4 \rightarrow (x=2 \lor x= -2)]$.
This sentence can be true or false.
At this stage, you still have a material conditional.
- Third, you manage to prove the consequent under the antecedent taken as hypothesis. It means that the consequent logically follows from the antecedent.
You can express this as:
$x^2 = 4 \implies ( x=2 \lor x= -2) $$\space$( $\space$ for all $x$)
meaning that there is no possible case in which $x^2 = 4$ is true and $( x=2 \lor x= -2)$ is false.
- In equation solving, we want the next step to follow logically from the previous one, so the arrrow is understood as a material conditional that holds in all possible cases , that is as " implies " or " $\implies$, or " logical implication".