How to show that the logical equivalence relation, $\equiv$, is an equivalence relation on $F$

Question

Let $F$ denote the set of well-formed formulas over a set Prop of propositional variables. Show that the logical equivalence relation, $\equiv$, is an equivalence relation on $F$.

if I assume $\equiv$ is a relation $E$, then I can prove that for any well-formed formulas $P$, $(P,P) \in E$ because $P \equiv P$, so $E$ is reflexive.

Is that a correct way to show reflexivity? And how should I show the Symmetry and Transitivity of $\equiv$, can anyone give me some advice?

Answer 1

To paraphrase the Wikipedia definition of logical equivalence (https://en.wikipedia.org/wiki/Logical_equivalence)statements: statements p and q are said to be logically equivalent if they are provable from each other under a set of axioms, or have the same truth value in every model.

• Reflexivity – Since p always has the same truth value as p
• Symmetry – Since each statement is provable from the other
• Transitivity – Since the application of successive proofs will provide the needed implication in one direction and symmetry will allow for the required proof in the opposite direction.