Equivalences are Interderivable/Proof 2

Theorem

If two propositional formulas are interderivable, they are equivalent:

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


Proof

Let $v$ be an arbitrary interpretation.

Then by definition of interderivable:

$\map v {p \iff q}$ if and only if $\map v p = \map v q$

Since $v$ is arbitrary, $\map v p = \map v q$ holds in all interpretations.

That is:

$p \dashv \vdash q$

$\blacksquare$


Sources

  • 1993: M. Ben-Ari: Mathematical Logic for Computer Science ... (previous) ... (next): Chapter $2$: Propositional Calculus: $\S 2.4$: Logical equivalence and substitution: Theorem $2.4.4$
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