Hurley, "...Logic" 8th ed. section 7.5 introduces conditional proof. Some exercises are designed to show proofs that are much easier using conditional proof. For example 7.5 I (2):
- $(F \implies E) \land (F \land E \implies R) $ Premise
- $(F \implies E) $ 1.
- $(F \land E \implies R)$ 1.
- $F$ Assumption
- $E$ 2.
- $F \land E$ 4, 5.
- $R$ 3.
- $F \implies R$ 4, 7.
I would like to see a proof of this without conditional proof. The allowable rules are these: Modus ponens, Modus tollens, Hypothetical syllogism, Disjunctive syllogism, Constructive Dilemma, And-introduction and elimination (named differently), Or-introduction, DeMorgan's laws, Commutivity, Associativity, distribution, double negation, Transposition (contrapositive), Material implication, Exportation $ (P \land Q) \implies R \iff (P \implies (Q \implies R))$, and tautologies $p \iff p \land p$, and $p \iff p \lor p$.
Translating things using material implication and then reducing using DeMorgan's and distribution is not working out for me. Constructive dilemma could be used on $ \lnot E \implies \lnot F $ and $E \implies (\lnot F \lor R))$ if we could introduce $E \lor \lnot E$, but there is no rule allowing this.