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Theorem mtpor 1692
 Description: Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1693, 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 alternative 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.)
Hypotheses
Ref Expression
mtpor.min ¬ 𝜑
mtpor.max (𝜑𝜓)
Assertion
Ref Expression
mtpor 𝜓

Proof of Theorem mtpor
StepHypRef Expression
1 mtpor.min . 2 ¬ 𝜑
2 mtpor.max . . 3 (𝜑𝜓)
32ori 390 . 2 𝜑𝜓)
41, 3ax-mp 5 1 𝜓
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 383 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 197  df-or 385 This theorem is referenced by:  mtpxor  1693  tfrlem14  7439  cardom  8764  unialeph  8876  brdom3  9302  sinhalfpilem  24136  mof  32086  dvnprodlem3  39496
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