Hoare triple for assignment P{x/E} x:=E {P}

I am trying to understand Hoare logic presented at Wikipedia,
Hoare logic at Wikipedia
Apparently, if I understand correctly, a Hoare triple $$\{P\}~ C ~\{Q\}$$ means


if P just before C, then Q holds immediately after C, as long as C terminates. (A)

However, the assignment axiom schema seems to be interpreted in a different way:

$$\frac{}{\{P[x/E]\} ~~x:=E~~ \{P\}}$$

The wikipedia says:

The assignment axiom means that the truth of $\{P[x/E]\}$ is equivalent to the after-assignment truth of $\{P\}$. Thus were $\{P[x/E]\}$ true prior to the assignment, by the assignment axiom, then $\{P\}$ would be true subsequent to which. Conversely, were $\{P[x/E]\}$ false prior to the assignment statement, $\{P\}$ must then be false consequently.

I think the Hoare triple only affirms that if P[x/E] before x:=E, then P(x) holds after x:=E. It DOES NOT affirm, by its definition, that if P(x) holds after x:=E, then P[x/E] holds before x:=E.

My naive question is, how can $\{P[x/E]\}$ before the assignment can be equivalent to $\{P\}$ after the assignment? Does this contradict with point (A) at the beginning of my post?

Notice that what Wikipedia is saying is that


The assignment axiom means that the truth of $\{P[x/E]\}$ is equivalent to the after-assignment truth of $\{P\}$.

In other words, ($P$ holds after the execution of $x:= E$) if ($P[x/E]$ holds before the execution). This is equivalent to the definition $A$ you provided, which is generally a more intuitive definition for Hoare triple.

StackExchange Computer Science Q#1864, answer score: 6