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Theorem mtp-or 1538
Description: Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtp-xor 1536, 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 is not true, and or (or both) are true, then 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.)
Ref Expression
Ref Expression

Proof of Theorem mtp-or
StepHypRef Expression
1 mtp-or.1 . 2
2 mtp-or.2 . . 3
32ori 364 . 2
41, 3ax-mp 8 1
Colors of variables: wff setvar class
Syntax hints:   wn 3   wo 357
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-or 359
This theorem is referenced by: (None)
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