How to translate this asserted symbolic tautology into natural language On https://de.wikipedia.org/wiki/Pr%C3%A4nexform a set of slides (lecture material in German) is cited. On slide 6 the following tautology is written:
$(\exists x B \rightarrow A) \leftrightarrow \forall x (B \rightarrow A) \quad x \notin \mathrm{Free}(A) $
I have difficulties to understand the meaning of this.
Currently I read

*

*$\exists x B \rightarrow A$: There is at least one $x$ such that $B \rightarrow A$.

*$\forall x (B \rightarrow A)$: The implication $B \rightarrow A$ holds for all $x$.

In my understanding the second assertion is much more general. Couldnt it be the case that there is only one $x$ (say $x=0$) such that B implies A?
Example: B: "$y$ is a polynomial of degree $x$", A: "$y$ is a constant number".
Where is my fallacy?


Example: B: "y is a polynomial of degree x", A: "y is a constant number".

Let's stipulate that the degree of a polynomial is a natural number.
Then $(\exists x.Bx) \to A$ becomes:
$$(B(0) \lor B(1) \lor B(2) \dots) \to A$$
Which by definition of B is $\text{true} \to A$, which is just $A$.
Also, $\forall x . (Bx \to A)$ becomes
$$(B(0) \to A) \land (B(1) \to A) \land (B(2) \to A) \dots$$
which becomes
$$\underbrace{(\text{true} \to A)}_\text{One of these}
\land \underbrace{(\text{false} \to A) \land (\text{false} \to A) \dots}_{\infty \text{ of these}}$$
Which again simplifies to just $A$.
