Theorem mtpxor 1696

Description: Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1695, one of the five "indemonstrables" in Stoic logic. The rule says, "if ph is not true, and either ph or ps (exclusively) are true, then ps must be true." Today the name "modus tollendo ponens" often refers to a variant, the inclusive-or version as defined in mtpor 1695. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1694, that is, it is exclusive-or df-xor 1465), rule 3 of [Sanford] p. 39 (where it is not as clearly stated which kind of "or" is used but it appears to be in the same sense as mptxor 1694), and rule A5 in [Hitchcock] p. 5 (exclusive-or is expressly used). (Contributed by David A. Wheeler, 4-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.) (Proof shortened by BJ, 19-Apr-2019.)

Hypotheses

Ref	Expression
mtpxor.min	|- -. ph
mtpxor.maj	|- ( ph \/_ ps )

Assertion

Ref	Expression
mtpxor	|- ps

Proof of Theorem mtpxor

Step	Hyp	Ref	Expression
1		mtpxor.min	. 2 |- -. ph
2		mtpxor.maj	. . 3 |- ( ph \/_ ps )
3		xoror 1471	. . 3 |- ( ( ph \/_ ps ) -> ( ph \/ ps ) )
4	2, 3	ax-mp 5	. 2 |- ( ph \/ ps )
5	1, 4	mtpor 1695	1 |- ps

Syntax hints:   -. wn 3   \/ wo 383   \/_ wxo 1464

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 197  df-or 385  df-an 386  df-xor 1465

This theorem is referenced by: (None)