  In this paper we recall some results for conditional events, compound
conditionals, conditional random quantities, p-consistency, and p-entailment.
Then, we show the equivalence between bets on conditionals and conditional
bets, by reviewing de Finetti's trivalent analysis of conditionals. But our
approach goes beyond de Finetti's early trivalent logical analysis and is based
on his later ideas, aiming to take his proposals to a higher level. We examine
two recent articles that explore trivalent logics for conditionals and their
definitions of logical validity and compare them with our approach to compound
conditionals. We prove a Probabilistic Deduction Theorem for conditional
events. After that, we study some probabilistic deduction theorems, by
presenting several examples. We focus on iterated conditionals and the
invalidity of the Import-Export principle in the light of our Probabilistic
Deduction Theorem. We use the inference from a disjunction, "$A$ or $B$", to
the conditional,"if not-$A$ then $B$", as an example to show the invalidity of
the Import-Export principle. We also introduce a General Import-Export
principle and we illustrate it by examining some p-valid inference rules of
System P. Finally, we briefly discuss some related work relevant to AI.
