Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  con1biimdc GIF version

Theorem con1biimdc 767
 Description: Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.)
Assertion
Ref Expression
con1biimdc (DECID 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))

Proof of Theorem con1biimdc
StepHypRef Expression
1 bi1 111 . . 3 ((¬ 𝜑𝜓) → (¬ 𝜑𝜓))
2 con1dc 753 . . 3 (DECID 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
31, 2syl5 28 . 2 (DECID 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
4 bi2 121 . . . 4 ((¬ 𝜑𝜓) → (𝜓 → ¬ 𝜑))
54con2d 554 . . 3 ((¬ 𝜑𝜓) → (𝜑 → ¬ 𝜓))
65a1i 9 . 2 (DECID 𝜑 → ((¬ 𝜑𝜓) → (𝜑 → ¬ 𝜓)))
73, 6impbidd 118 1 (DECID 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  DECID wdc 742 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-dc 743 This theorem is referenced by:  con1bidc  768  con1biddc  770
 Copyright terms: Public domain W3C validator