Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  con1bidc GIF version

Theorem con1bidc 859
 Description: Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
Assertion
Ref Expression
con1bidc (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑))))

Proof of Theorem con1bidc
StepHypRef Expression
1 con1biimdc 858 . . . 4 (DECID 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
21adantr 274 . . 3 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
3 con1biimdc 858 . . . 4 (DECID 𝜓 → ((¬ 𝜓𝜑) → (¬ 𝜑𝜓)))
43adantl 275 . . 3 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜓𝜑) → (¬ 𝜑𝜓)))
52, 4impbid 128 . 2 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑)))
65ex 114 1 (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 819 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 603  ax-in2 604  ax-io 698 This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820 This theorem is referenced by:  con2bidc  860
 Copyright terms: Public domain W3C validator