Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  necon2bbiddc GIF version

Theorem necon2bbiddc 2391
 Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon2bbiddc.1 (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓𝐴𝐵)))
Assertion
Ref Expression
necon2bbiddc (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))

Proof of Theorem necon2bbiddc
StepHypRef Expression
1 necon2bbiddc.1 . . . 4 (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓𝐴𝐵)))
2 bicom 139 . . . 4 ((𝜓𝐴𝐵) ↔ (𝐴𝐵𝜓))
31, 2syl6ib 160 . . 3 (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵𝜓)))
43necon1bbiddc 2387 . 2 (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓𝐴 = 𝐵)))
5 bicom 139 . 2 ((¬ 𝜓𝐴 = 𝐵) ↔ (𝐴 = 𝐵 ↔ ¬ 𝜓))
64, 5syl6ib 160 1 (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  DECID wdc 820   = wceq 1332   ≠ wne 2324 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 604  ax-in2 605  ax-io 699 This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-ne 2325 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator