Relevant implication and the weak deduction theorem (Q1103605)

Relevant implicational calculus \(R_{\to}\) is the weakest system closed under modus ponens, substitution and relevant deduction: if there is a deduction of B from the set \(X\cup \{A\}\), where A is actually used, then \(A\to B\) is deducible from X. The author shows that this is no longer so, if one replaces relevant deduction with what he calls weak deduction theorem: if \(X\cup \{A\}\vdash B\) and \(X\nvdash B\), then \(X\vdash A\to B\).