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Theorem mtpor 1733
Description: Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1734, 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 847 . 2 𝜑𝜓)
41, 3ax-mp 5 1 𝜓
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 199  df-or 834
This theorem is referenced by:  mtpxor  1734  tfrlem14  7833  cardom  9211  unialeph  9323  brdom3  9750  sinhalfpilem  24755  mofal  33278  dvnprodlem3  41664
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