Theorem mtpor 1425

Description: Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1426, 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 723	. 2 |- ( -. ph -> ps )
4	1, 3	ax-mp 5	1 |- ps

Syntax hints:   -. wn 3   \/ wo 708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  mtpxor  1426  ordtriexmid  4522