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Theorem mtpor 1655
Description: Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1656, 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  ph is not true, and  ph or  ps (or both) are true, then  ps 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  |-  -.  ph
mtpor.max  |-  ( ph  \/  ps )
Assertion
Ref Expression
mtpor  |-  ps

Proof of Theorem mtpor
StepHypRef Expression
1 mtpor.min . 2  |-  -.  ph
2 mtpor.max . . 3  |-  ( ph  \/  ps )
32ori 377 . 2  |-  ( -. 
ph  ->  ps )
41, 3ax-mp 5 1  |-  ps
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 370
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 189  df-or 372
This theorem is referenced by:  mtpxor  1656  tfrlem14  7114  cardom  8425  unialeph  8537  brdom3  8961  sinhalfpilem  23430  mof  31082  dvnprodlem3  37833
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