Theorem mtpxor 1437

Description: Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1436, 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 1436. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1435, that is, it is exclusive-or df-xor 1387), 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 1435), 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 1390	. . 3 |- ( ( ph \/_ ps ) -> ( ph \/ ps ) )
4	2, 3	ax-mp 5	. 2 |- ( ph \/ ps )
5	1, 4	mtpor 1436	1 |- ps

Syntax hints:   -. wn 3   \/ wo 709   \/_ wxo 1386

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 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-xor 1387

This theorem is referenced by: (None)