Mathematical Logic Inference theory

Question

Show that $(L \lor m)$ logically follows from:

• $p\land q\land r$
• $(q\leftrightarrow r) \to (L \lor m)$

how to solve this using inference theory? i could get till here,

1. p,q,r --- (rule p)

2. (q → r) ∧ (r → q) → (L ∨ m) ---(rule p)

3. (q → r),(r → q) → (L ∨ m)

what to do after this.. i got stuck here..

Answer 1

Hint

From $p \land q \land r$ derive $q$ and $r$ separately using Simplification rule.

From $q$ derive $\lnot r \lor q$ by Addition and transform it into the equivalent $(r \to q)$.

In the same way, from $r$ derive $(q \to r)$.

Then use Conjunction to get $(r \to q) \land (q \to r)$ followed by Biconditional introduction to get :

$(q \leftrightarrow r)$.