I want to show that a statement with several quantifiers, e.g., "$f(a, b) <= 0$ for all $a\in [0, 3]$ and all $b\in [-\infty, -1]$", is equivalent to another statement with several quantifiers, e.g., "$g(a, b) <= 0$ for all $a \in[-10, 15]$ and $b < 0$", where $f$ and $g$ are two functions from $\mathbb{R}^2 \rightarrow \mathbb{R}$, although this is irrelevant to my question.
When it comes to writing that they are equivalent and, more specifically, when referencing the first statement, I'm not sure how to do it.
Which option of the two below is most frequent in mathematical writing?
OPTION 1
Line 1: We have shown that
Line 2: $f(a,b) <=0$ Eq.(1)
Line 3: for all $a\in[0,3]$ and all $b\in[-\infty,-1]$.
Line 4: Now we will prove Eq.(1) is equivalent to the condition $g(a,b)<=0$ for all $a\in[-10,15]$ and $b<0$.
Eq.(1) above refers just to “$f(a,b)<=0$" and doesn’t include the quantifiers “for all $a\in[0,3]$ and all $b\in[-\infty,-1]$”. Therefore, I wonder if Line 4 can be interpreted as “Now we will prove $f(a,b)<=0$ is equivalent to the condition $g(a,b)<=0$ for all $a\in[-10,15]$ and $b<0$”, which clearly is not what I want to show.
OPTION 2
To address this potential issue, I guess the solution would be as follows:
Line 1: We have shown that
Line 2: $f(a, b) <= 0$ for all $a\in [0, 3]$ and all $b\in [-\infty, -1]$. Eq.(1)
Line 3: Now we will prove Eq.(1) is equivalent to the condition $g(a,b)<=0$ for all $a\in[-10,15]$ and $b<0$.
Any help would be much appreciated. Also, I wonder whether I should write "statement" instead of "condition" in Line 4.
Style will guide writers on such issues. This is a Question about math writing & you can check out Style guides , eg Knuth or American Mathematical Society or other alternatives.
Here is how I might have written it :
We have shown Proposition P1 :
$\forall a \in [0,3] , \forall b \in [-\infty,-1] : f(a,b) \leq 0 \tag{P1}$
Now we will prove P1 is equivalent to Proposition P2 :
$\forall a \in [-10,15] , b \le 0 : g(a,b) \leq 0 \tag{P2}$
High-light :
Use Math Symbols like $\forall$ & $\exists$ rather than "for all" & "there exists"
Use Math Symbols like $\leq$ & $\geq$ rather than "<=" & ">="
Prefer words like Corollary & Proposition & Lemma & Theorem for Conclusions & use words like Condition & Criteria within those Conclusions.
Use "tags" to refer later.
UPDATE :
When we do not want to make Propositions & it is in the flow of some argument , then this tweak can work :
We have shown that :
$\forall a \in [0,3] , \forall b \in [-\infty,-1] : f(a,b) \leq 0 \tag{7}$
Now we will prove that (7) is equivalent to the following :
$\forall a \in [-10,15] , b \le 0 : g(a,b) \leq 0 \tag{8}$
We use (8) to get our Core result (9) like this :
$\cdots$