Definition:Provable Equivalence

Definition
Let $\mathcal P$ be a proof system for a formal language $\mathcal L$.

Let $\phi, \psi$ be $\mathcal L$-WFFs.

Then $\phi$ and $\psi$ are $\mathscr P$-provably equivalent :


 * $\phi \vdash_{\mathscr P} \psi$ and $\psi \vdash_{\mathscr P} \phi$

that is, they are $\mathscr P$-provable consequences of one another.

The provable equivalence of $\phi$ and $\psi$ can be denoted by:


 * $\phi \dashv \vdash_{\mathscr P} \psi$

Also see

 * Definition:Provable Consequence


 * Definition:Logical Equivalence
 * Definition:Semantic Equivalence