Propositional Logic - Exercise Hurley Chapter 7.3

Question

I'm trying to solve this exercise

Exercise
1. $E \lor \lnot (D \lor C)$
2. $(E \lor \lnot D) \to C$

$\quad\therefore E$

I tried and I went to 11 steps but I'm stuck, I tried De Morgans, Transposition, Exportation and Material implication but I can't get to the E conclusion.

Thank you.

Answer 1

HINT: Try a proof by contradiction (your system may call it an Indirect Proof): Assume $\neg E$, and see if you can derive a contradiction.

Answer 2

I tried De Morgans, Transposition, Exportation and Material implication but I can't get to the E conclusion.

1. $\rm E∨¬(D∨C)$ as premised
2. $\rm (E∨¬D)→C$ as premised
3. $\rm E\lor(\lnot D\land\lnot C)$ by de Morgan's of 1
4. $\rm \lnot (E\lor\lnot D)\lor C$ by Implication Equivalence on 2
5. $\rm (\lnot E\land D)\lor C$ by de Morgan's on 4
6. $\rm (E\lor(\lnot D\land\lnot C))\land((\lnot E\land D)\lor C)$ be conjunction of 3 and 5
7. distribution and other stuff happens...