 Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  con1bdc GIF version

Theorem con1bdc 813
 Description: Contraposition. Bidirectional version of con1dc 794. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
con1bdc (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑))))

Proof of Theorem con1bdc
StepHypRef Expression
1 con1dc 794 . . . 4 (DECID 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
21adantr 271 . . 3 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
3 con1dc 794 . . . 4 (DECID 𝜓 → ((¬ 𝜓𝜑) → (¬ 𝜑𝜓)))
43adantl 272 . . 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 783 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668 This theorem depends on definitions:  df-bi 116  df-dc 784 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator