MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mtp-xor Unicode version

Theorem mtp-xor 1525
Description: Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, 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 mtp-or 1527. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mpto2 1524, that is, it is exclusive-or df-xor 1296), 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 mpto2 1524), 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.)
Hypotheses
Ref Expression
mtp-xor.1  |-  -.  ph
mtp-xor.2  |-  ( ph  \/_ 
ps )
Assertion
Ref Expression
mtp-xor  |-  ps

Proof of Theorem mtp-xor
StepHypRef Expression
1 mtp-xor.1 . . 3  |-  -.  ph
2 mtp-xor.2 . . . 4  |-  ( ph  \/_ 
ps )
3 xorneg 1304 . . . 4  |-  ( ( -.  ph  \/_  -.  ps ) 
<->  ( ph  \/_  ps ) )
42, 3mpbir 200 . . 3  |-  ( -. 
ph  \/_  -.  ps )
51, 4mpto2 1524 . 2  |-  -.  -.  ps
65notnotri 106 1  |-  ps
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/_ wxo 1295
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-xor 1296
  Copyright terms: Public domain W3C validator