Theorem mtpor 1420

Description: Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1421, 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 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.)

Hypotheses

Ref	Expression
mtpor.min	|- -. ph
mtpor.max	|- ( ph \/ ps )

Assertion

Ref	Expression
mtpor	|- ps

Proof of Theorem mtpor

Step	Hyp	Ref	Expression
1		mtpor.min	. 2 |- -. ph
2		mtpor.max	. . 3 |- ( ph \/ ps )
3	2	ori 718	. 2 |- ( -. ph -> ps )
4	1, 3	ax-mp 5	1 |- ps

Syntax hints:   -. wn 3   \/ wo 703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  mtpxor  1421  ordtriexmid  4505