Theorem mtpxor 1416

Description: Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1415, one of the five "indemonstrables" in Stoic logic. The rule says, "if 𝜑 is not true, and either 𝜑 or 𝜓 (exclusively) are true, then 𝜓 must be true". Today the name "modus tollendo ponens" often refers to a variant, the inclusive-or version as defined in mtpor 1415. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1414, that is, it is exclusive-or df-xor 1366), 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 1414), 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	⊢ ¬ 𝜑
mtpxor.maj	⊢ (𝜑 ⊻ 𝜓)

Assertion

Ref	Expression
mtpxor	⊢ 𝜓

Proof of Theorem mtpxor

Step	Hyp	Ref	Expression
1		mtpxor.min	. 2 ⊢ ¬ 𝜑
2		mtpxor.maj	. . 3 ⊢ (𝜑 ⊻ 𝜓)
3		xoror 1369	. . 3 ⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ∨ 𝜓))
4	2, 3	ax-mp 5	. 2 ⊢ (𝜑 ∨ 𝜓)
5	1, 4	mtpor 1415	1 ⊢ 𝜓

Syntax hints:  ¬ wn 3   ∨ wo 698   ⊻ wxo 1365

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 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-xor 1366

This theorem is referenced by: (None)