Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  mtpxor GIF version

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 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 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 ¬ 𝜑
mtpxor.maj (𝜑𝜓)
Assertion
Ref Expression
mtpxor 𝜓

Proof of Theorem mtpxor
StepHypRef Expression
1 mtpxor.min . 2 ¬ 𝜑
2 mtpxor.maj . . 3 (𝜑𝜓)
3 xoror 1270 . . 3 ((𝜑𝜓) → (𝜑𝜓))
42, 3ax-mp 7 . 2 (𝜑𝜓)
51, 4mtpor 1316 1 𝜓
 Colors of variables: wff set class 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)
 Copyright terms: Public domain W3C validator