I found this phrase in the page 60 of the book "A Transition to Advanced Mathematics, 8th Edition, written by Smith/Eggen/St. Andre."
"Proving that ∼(∃x) P (x), is false is equivalent to proving that (∀x )∼P (x) is true."
It seems incorrect to me, but as I am a student, I guess that I am wrong.
My reasoning is that if we work with the equation of the right side:
(∀x)∼P(x) is true, we can see it as
("(∀x)∼P (x) = true") and this would be equal to
("∼ ((∀x)∼P (x)) = ∼(true)")=
(∃x)P(x) = false.
then (∃x)P(x) is false.
That is the opposite of our goal.
Am I wrong?