Rule of Material Equivalence
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Theorem
The rule of material equivalence 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
- $\vdash \left({p \iff q}\right) \iff \left({\left({p \implies q}\right) \land \left({q \implies p}\right)}\right)$
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
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $3.2$: The Rule of Replacement