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Theorem mtpor 1357
Description: Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1358, 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 𝜑 is not true, and 𝜑 or 𝜓 (or both) are true, then 𝜓 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
mtpor.min ¬ 𝜑
mtpor.max (𝜑𝜓)
Ref Expression
mtpor 𝜓

Proof of Theorem mtpor
StepHypRef Expression
1 mtpor.min . 2 ¬ 𝜑
2 mtpor.max . . 3 (𝜑𝜓)
32ori 675 . 2 𝜑𝜓)
41, 3ax-mp 7 1 𝜓
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wo 662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  mtpxor  1358  ordtriexmid  4273
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