Theorem mtpxor 1317

Description: Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1316, 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 1316. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1315, that is, it is exclusive-or df-xor 1267), 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 1315), 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 1270	. . 3 |- ( ( ph \/_ ps ) -> ( ph \/ ps ) )
4	2, 3	ax-mp 7	. 2 |- ( ph \/ ps )
5	1, 4	mtpor 1316	1 |- ps

Syntax hints:   -. wn 3   \/ wo 629   \/_ wxo 1266

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-xor 1267

This theorem is referenced by: (None)