# Formal Proof using replacement rules and inference laws

Justify each step of the formal proof using inference laws and replacement rules. We know that $(p \to\ q) \land\ (p \to\ r) = p \to\ (q \land\ r)$

$1.$ $p \to\ q$ $\mathbf{(premise)}\$

$2.$ $q \to\ \overline{r}\$ $\mathbf{(premise)}\$

$3.$ $p \to\ r$ $\mathbf{(premise)}\$

$4.$ $\overline{q}\ \lor\ \overline{r}$ $\mathbf{(2. Implication )}\$

$5.$ $(\overline{q \land\ r})$ $\mathbf{(4. DeMorgan)}\$

$6.$ $(p \to\ q) \land\ (p \to\ r)$ $\mathbf{(?)}\$

$7.$ $p \to\ (q \land\ r)$ $\mathbf{(6. EquivalenceFromQuestion)}\$

$8.$ $\overline{p}\$ $\mathbf{(?)}\$

It's probably very obvious but I am struggling to fill in the steps that are represented by ?'s and am really not sure what I am missing. Any insight would be great thanks.