formal proof with natural deduction

Question

Hi I am looking to prove

$r\wedge q\Leftrightarrow r \vdash r\Rightarrow q$

using natural deduction

I get:

• $r\wedge q\Leftrightarrow r$, assumption
• $r\vdash q$
• $r$, assumption

I assume that I also need to formally prove $q$ but cant figure out how to do it(and what rule would I use?) Am I correct so far? thank you.

Answer 1

(1) Assume $r$.

Then from the premise $(r\land q) \longleftrightarrow r$, which means $((r\land q)\rightarrow r)\land (r\rightarrow (r\land q))$, we have

(2) $r\rightarrow (r\land q)$,

Since $r$ is assumed, we have, by modus ponens using $(1), (2)$, we have

(3) $r\land q$.

This gives us, from (3),

(4) $q$.

From the assumption of $r$, we've deduced $q$.

Therefore, $r\rightarrow q$.