Theorem mtpor 1356

Description: Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1357, 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 674	. 2 |- ( -. ph -> ps )
4	1, 3	ax-mp 7	1 |- ps

Syntax hints:   -. wn 3   \/ wo 661

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  mtpxor  1357  ordtriexmid  4265