Theorem mtpxor 1470

Description: Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1469, 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 1469. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1468, that is, it is exclusive-or df-xor 1420), 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 1468), 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 1423	. . 3 |- ( ( ph \/_ ps ) -> ( ph \/ ps ) )
4	2, 3	ax-mp 5	. 2 |- ( ph \/ ps )
5	1, 4	mtpor 1469	1 |- ps

Syntax hints:   -. wn 3   \/ wo 715   \/_ wxo 1419

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-in1 619  ax-in2 620  ax-io 716

This theorem depends on definitions:  df-bi 117  df-xor 1420

This theorem is referenced by: (None)