Theorem mtpxor 1404

Description: Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1403, 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 1403. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1402, that is, it is exclusive-or df-xor 1354), 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 1402), 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 1357	. . 3 |- ( ( ph \/_ ps ) -> ( ph \/ ps ) )
4	2, 3	ax-mp 5	. 2 |- ( ph \/ ps )
5	1, 4	mtpor 1403	1 |- ps

Syntax hints:   -. wn 3   \/ wo 697   \/_ wxo 1353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-xor 1354

This theorem is referenced by: (None)