Using truth tables for the six formulas p, q, p → q, q → p, p → p, and (p →q) → q, show that every formula using only p and q as propositional letters and containing only the implication connective must be logically equivalent to one of the six formulas.
I found this problem in Introduction to Logic by Paul Herrick. Currently, I'm thinking that I need to show that every single formula using only the implication connective can only have at most one false in the final column of the truth table. This would allow me to show that what I have in these six formulas is a closed system. But I'm not sure how to do this rigorously.
Any help would be great! Thanks in advance!