Question

definition of implication: why does P(x) v Q(x) become ~Q(x) -> P(x) rather than ~P(x) -> Q(x)?

Answer 1

The easiest way to prove equivalence is to draw out a truth table and then compare the values. I'm going to show a truth table using proposition logic, it's the same result as using predicate logic.
P(x) v Q(x)

P |Q || PvQ || ~Q->P <----Notice how this column matches the PvQ but if you were to
---|---||--------||---------- <----continue the truth table with ~P->Q it would not be equivalent
T T T T
T F T T
F T T T
F F F F

Let me know if you would like an example, if the truth table doesn't help.