Rule of Material Equivalence

Theorem

The is a valid deduction sequent in propositional logic:

If we can conclude that $p$ implies $q$ and if we can also conclude that $q$ implies $p$, then we may infer that $p$ if and only if $q$.


Formulation 1

$p \iff q \dashv \vdash \paren {p \implies q} \land \paren {q \implies p}$

Formulation 2

$\vdash \paren {p \iff q} \iff \paren {\paren {p \implies q} \land \paren {q \implies p} }$


Sources

  • 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S \text{II}.12$: Laws of sentential calculus
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): material equivalence
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): material equivalence
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