Help with deduction of logic proposition

Question

I need help with this propositional logic deduction..

$( S \vee \neg B) \wedge (T \vee \neg I)$

$B \wedge I$

$\therefore S \wedge T$

---

$B \wedge I \implies \neg B \vee \neg I$

but I get stuck here.

Answer 1

You're approaching this from the wrong direction. Start like this,

$B \wedge I$

$B \wedge I \implies B$

$\therefore B$

$B \wedge I$

$B \wedge I \implies I$

$\therefore I$

and then apply these two conclusions to your first premise.

Answer 2

The rules you need $$\begin{split}\phi\wedge \psi&\vdash \phi &\quad&\text{Conjunctive Elimination} \\\phi\vee \neg \psi, \psi&\vdash \phi &&\text{Disjunctive Sylogism}\\\phi,\psi&\vdash \psi\wedge \psi&&\text{Conjunctive Introduction}\end{split}$$