I have the following proposition:
If $a$, $b$, and $c$ are real numbers for which there is a real number $x \not = 0$ such that $ax^2 + bx + c = 0$, then $cx^2 + bx + a$ has a rational root.
Therefore, the hypothesis and conclusion are as follows:
Hypothesis: $a$, $b$, and $c$ are real numbers for which there is a real number $x \not = 0$ such that $ax^2 + bx + c = 0$.
Conclusion: $cx^2 + bx + a$ has a rational root.
I'm wondering if it is completely equivalent to reword the hypothesis as follows:
Hypothesis: For all real numbers $a$, $b$, and $c$, there is a real number $x \not = 0$ such that $ax^2 + bx + c = 0$.
I would greatly appreciate it if someone could please take the time to clarify my thoughts.