This is probably a little basic, but say we want to 'prove' that 'If $x + 1 = x(1 + a)$ then $ax = 1$.

Now, back in high school I'd have just gone for the line-by-line method, i.e.

\begin{align\*}
x + 1 &= x(1 + a)\\
x + 1 &= x + ax\\
1 &= ax
\end{align\*}

But now that I'm starting to write simple proofs it seems like the examples I'm given make more use of phrasing like '[...] $x + 1 = x + ax$ which implies that $1 = ax$ [...]' instead of the above. I'm not sure if it's just the style of this professor to write it that way, but are these two things equivalent? Is the line-by-line layout above just the same as saying $$x + 1 = x(1 + a) \iff x + 1 = x + ax \iff 1 = ax?$$ or perhaps is it the softer $$x + 1 = x(1 + a) \implies x + 1 = x + ax \implies 1 = ax?$$

Basically, what do the line breaks in the line-by-line method say about each statement that follows another? I wonder if it's just generally ambiguous and so to be avoided in favor of the specific implication terminology?