Biconditional Elimination

Theorem

The rule of biconditional elimination is a valid argument in types of logic dealing with conditionals $\implies$ and biconditionals $\iff$.

This includes classical propositional logic and predicate logic, and in particular natural deduction.

$(1): \quad$ If we can conclude $\phi \iff \psi$, then we may infer $\phi \implies \psi$.

$(2): \quad$ If we can conclude $\phi \iff \psi$, then we may infer $\psi \implies \phi$.

$\text {(1)}: \quad$

$\ds p \iff q$

$\vdash$

$\ds p \implies q$

$\text {(2)}: \quad$

$\ds p \iff q$

$\vdash$

$\ds q \implies p$

Some sources refer to the as the rule of Biconditional-Conditional.