Can two propositions be both tautologies but not logically equivalent?

Question

Hello this might sound like a play on word although I am kinda of confused. Can it be possible that two propositions are both tautologies but not logically equivalent?

Here is an example:

$(\neg p \wedge (p \vee q)) \rightarrow p$ and $(p \wedge(p\rightarrow q)) \rightarrow q$ now these two are both tautologies, and if I try to show that they are with identities they are both true at the end where:

$(p \vee T)$ and $(q \vee T)$ (Thus both true)

but when you try to show logical equivalence with identities, then you can't.

Is this right? and if yes why? How is this called?

Thank you

Answer 1

Any two tautologies are logically equivalent.

And note: $p \lor \top = q \lor \top = \top$

So they really are equivalent!