I'm not sure I understood the difference between
'If $p$ then $q$' ,i.e. $p \to q $
and
'$p$ implies $q$', i.e. $p \implies q $
in a right way.
I thought that 'if $p$ then $q$' is that you assume the $p$ is true, which means doesn't need to be proved.
However, '$p$ implies $q$' is that 'if $p$ is true then $q$ is also true'. Therefore, $p$ needs to be proved and if it is proved to be true then $q$ is true automatically.
Did I understand correctly?
$p \rightarrow q$ is a propositional logic expression that uses the material conditional or material implication: a truth-functional operator that determines the truth-value of the statement as a whole based on the truth-values of $p$ and $q$ themselves. Informally, this statement is expressed as 'if $p$ then $q$'
Some books and authors use $p \Rightarrow q$ to mean the same thing, i.e. they also use $\Rightarrow$ as the symbol for the material conditional.
However, many logicians use the $\Rightarrow$ for the meta-logical symbol for logical implication. Logical implication is really quite different from the material conditional. Where the material conditional takes two logic statements and combines them into one new logic statement, the logical equivalence is a statement about two different logic statements: We say that a statement $\varphi$ logically implies a statement $\psi$ if and only if there is no possible truth-assignment that sets $\varphi$ to true and $\psi$ to false. An example would be $\neg \neg p \Rightarrow p$.
Now, if $p$ and $q$ are atomic statements, then we do not have $p \Rightarrow q$: $p$ does not logically imply $q$, because we can set $p$ to true and $q$ to false. However, when describing a world or context, we can still use $p \rightarrow q$. For example, we can say that 'If there is smoke, then there is fire'. Using $p$ for 'there is smoke', and $q$ for 'there is fire', we can write this statement as $p \rightarrow q$, and we would say that in the world we are trying to describe, $p \rightarrow q$ is true. However, this does not make $p \Rightarrow q$ true. In fact, $p \Rightarrow q$ is false, since we can imagine a world/context where there is smoke, but no fire.
There is no difference between " If $p$ then $q$" and "$p$ implies $q$"
$p$ implies $q$ sounds more formal so its used more often in formal proofs.
You do not have to prove that $p$ is true, you assume $p$ is true and show that $q$ is true.
"if $p$ then $q$" and "$p$ implies $q$" are logically equivalent--even in instances where $p$ is never true. [Although as far as actual usage in papers and textbooks goes, the latter tends to be used more for instances when $p$ has already been established i.e., "We just established $p$, and as $p$ implies $q$, we just established $q$ as well", while the former tends to be used more when neither $p$ nor $q$ have yet been establihsed--"if $p$ is true then so is $q$, so to prove $q$ we will next establish $p$ and then $q$ will follow".]
In either case, if $p$ is true, then so is $q$.
There is no formal difference between the two, they're just different words meaning the same thing.
Sometimes people will write proofs starting with "If p, then..." where the proceeding logic assumes p. But they still haven't proven p, and all of the conclusions are dependent on the still unproved p.
