Can more than one conditional proof be used in an argument?

**Briefly yes, it is permissible to use C.P. more than once in a given formal proof. But either one must be “nested” fully within the other, or they must be completely distinct. They cannot “overlap.” Here is an example for an interesting argument:**

- (P ∨ Q) ⊃ R / ∴ (P ⊃ R) • (Q ⊃ R)
- P C.P.A.
- P ∨ Q 2 Add.
- R 1, 3 M.P.
- P ⊃ R 2-4 C. P.
- Q C.P.A.
- Q ∨ P 6 Add.
- P ∨ Q 7 Com.
- R 1, 8 M.P.
- Q ⊃ R 6-9 C.P.
- (P ⊃ R) • (Q ⊃ R) 5, 10 Conj.

Q.E.D.