> The ⊢ symbol has not changed; it means that the formula to which it applies is asserted to be true. ⊃ is logical implication, and ≡ is logical equivalence.
A strange thing happened to me in mathematics. When I got to the point where these symbols started showing up (ninth grade, more or less) I did not get a thorough explanation of the symbols; they just appeared and I tried to intuit what they meant. As more symbols crept into my math, I tried to ignore them where possible. Eventually this meant that I could not continue learning math, as it became mostly all such symbols.
I got as far as a minor in math. I'm not sure how any of this this happened, but I wish I had a table of these symbols in ninth grade.