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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

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Any two tautologies are logically equivalent.

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

So they really are equivalent!

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    But then all tautologies are logically equivalent between each other? – 2017-02-04
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    Correct!! You can rewrite any tautology as $\top$. – 2017-02-04
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    Ah ok, now it makes sense, thanks! – 2017-02-04
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    @Bram28 If all tautologies are logically equivalent; then why do we have different axioms that are tautologies? See http://math.stackexchange.com/questions/2119044/changing-one-axiom-of-hpc#comment4360933_2119044 – 2017-02-05
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    @user400188 Excellent question!! The answer is that for these kinds of axiom systems we don't have the rules of equivalence available to us to actually derive one from the others ... In fact, those systems typically have just one rule of inference: Modus Ponens. But yes, if we did have the equivalence rules available to use you would be absolutely right: we would need just one axiom instead of a bunch! – 2017-02-05
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    One very syntactic way of looking at this is consider the set of well-formed formulas. We then generate an equivalence relation, in fact a [congruence](https://en.wikipedia.org/wiki/Congruence_relation) with respect to the logical connectives, by saying for each formula $F$ which is declared an axiom, $F\sim\top$. Theorems are then the elements of the equivalence class of $\top$. New axioms make this equivalence class (potentially) larger. (You can then think of studying the (often algebraic) structure of the well-formed formulas quotiented by this congruence.) – 2017-02-05