Theorem mtpor

Description: Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor , one of the five original "indemonstrables" in Stoic logic. However, in Stoic logic this rule used exclusive-or, while the name modus tollendo ponens often refers to a variant of the rule that uses inclusive-or instead. The rule says, "if ph is not true, and ph or ps (or both) are true, then ps must be true". An alternate phrasing is: "once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth". -- Sherlock Holmes (Sir Arthur Conan Doyle, 1890: The Sign of the Four, ch. 6). (Contributed by David A. Wheeler, 3-Jul-2016) (Proof shortened by Wolf Lammen, 11-Nov-2017)

	Ref	Expression
Hypotheses	mtpor.min	$$⊢\neg \phi$$
	mtpor.max	$$⊢\left(\phi \vee \psi \right)$$
Assertion	mtpor	$$⊢\psi$$

Proof

Step	Hyp	Ref	Expression
1		mtpor.min	$$⊢\neg \phi$$
2		mtpor.max	$$⊢\left(\phi \vee \psi \right)$$
3	2	ori	$$⊢\neg \phi \to \psi$$
4	1 3	ax-mp	$$⊢\psi$$