Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  mtp-or Structured version   GIF version

Theorem mtp-or 1314
 Description: Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtp-xor 1313, 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
mtp-or.1 ¬ φ
mtp-or.2 (φ ψ)
Assertion
Ref Expression
mtp-or ψ

Proof of Theorem mtp-or
StepHypRef Expression
1 mtp-or.1 . 2 ¬ φ
2 mtp-or.2 . . 3 (φ ψ)
32ori 641 . 2 φψ)
41, 3ax-mp 7 1 ψ
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∨ wo 628 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 629 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  ordtriexmid  4210
 Copyright terms: Public domain W3C validator