Theorem mtp-or 1538

Description: Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtp-xor 1536, 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
mtp-or.1	|- -. ph
mtp-or.2	|- (ph \/ ps)

Assertion

Ref	Expression
mtp-or	|- ps

Proof of Theorem mtp-or

Step	Hyp	Ref	Expression
1		mtp-or.1	. 2 |- -. ph
2		mtp-or.2	. . 3 |- (ph \/ ps)
3	2	ori 364	. 2 |- (-. ph -> ps)
4	1, 3	ax-mp 5	1 |- ps

Syntax hints:  -. wn 3   \/ wo 357

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 177  df-or 359

This theorem is referenced by: (None)