Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  mpto1 Structured version   GIF version

Theorem mpto1 1311
 Description: Modus ponendo tollens 1, one of the "indemonstrables" in Stoic logic. See rule 1 on [Lopez-Astorga] p. 12 , rule 1 on [Sanford] p. 40, and rule A3 in [Hitchcock] p. 5. Sanford describes this rule second (after mpto2 1312) as a "safer, and these days much more common" version of modus ponendo tollens because it avoids confusion between inclusive-or and exclusive-or. (Contributed by David A. Wheeler, 2-Mar-2018.)
Hypotheses
Ref Expression
mpto1.1 φ
mpto1.2 ¬ (φ ψ)
Assertion
Ref Expression
mpto1 ¬ ψ

Proof of Theorem mpto1
StepHypRef Expression
1 mpto1.1 . 2 φ
2 mpto1.2 . . 3 ¬ (φ ψ)
3 imnan 623 . . 3 ((φ → ¬ ψ) ↔ ¬ (φ ψ))
42, 3mpbir 134 . 2 (φ → ¬ ψ)
51, 4ax-mp 7 1 ¬ ψ
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97 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 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  mpto2  1312
 Copyright terms: Public domain W3C validator