# Simplifying Sentences that Precede a Conditional

Suppose we have the following proposition:

Suppose that $$x \in \mathbb Z$$, that $$a$$ is even, and that $$b$$ is odd. If $$x^2-ax+b$$ is even, then $$x$$ is odd.

I am not interested in proving the proposition; I am more curious about the structure of it. Namely, suppose I present a second, similar proposition:

If $$x \in \mathbb Z$$, that $$a$$ is even, $$b$$ is odd, and $$x^2-ax+b$$ is even, then $$x$$ is odd.

Are these two propositions the same? If not, why? If they are the same, which is more "proper"? Is there a particular reason one would prefer one over the other? I know this is likely trivial, but I cannot seem to find answers about this issue elsewhere.

• This is just currying. Commented Jul 20 at 21:23
• They are logically equivalent, so what matters here is the ease of comprehension. I prefer the first version, but that is just my opinion. (BTW: in your second version, the word "that" is a mistake.) Commented Jul 20 at 21:26

$$P \to (Q \to R) \Leftrightarrow (P \land Q) \to R$$