TITLE: "All true theorems are logically equivalent"
QUESTION [9 upvotes]: I've seen the phrase "all true theorems are logically equivalent" thrown around here, when people ask if a theorem X and a theorem Y are logically equivalent.
What is meant by this? Are they just referring to the fact that an implication with a true consequent is always a true statement, therefore the biconditional $P \Longleftrightarrow Q$ is true for any true $P$, $Q$, or do they mean something more meaningful?

REPLY [2 votes]: In some cases, when someone says "$P$ and $Q$ are equivalent", they mean more than just "$P \Longleftrightarrow Q$ is provable".  They mean that a proof of $P \Longleftrightarrow Q$ is much easier than a proof of $P$ or a proof of $Q$.  
The motto "All true theorems are logically equivalent" is in opposition to this.