Give an equational proof $ \vdash p \land (q \equiv p) \equiv p \land q $

Question

Give an equational proof of $$ \vdash p \land (q \equiv p) \equiv p \land q $$

How can I give equational proof for this formula ?

See George Tourlakis, Mathematical Logic (2008) or this post for a list of axioms and theorems.

Answer 1

Hint.

Start with :

$(q \equiv p) \equiv (q \rightarrow p) \land (p \rightarrow q)$ --- (2.4.26)

and then :

$(q \equiv p) \equiv (\lnot q \lor p) \land (\lnot p \lor q)$ --- by (2.4.11)

to obtain :

$[p \land (q \equiv p)] \equiv p \land [(\lnot q \lor p) \land (\lnot p \lor q)]$.

Then apply Associativity of $\land$ to the RHS and go on with the usual transformations, according to the rules you have in your list, until you reach the goal.