Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  necon2abiddc GIF version

Theorem necon2abiddc 2374
 Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon2abiddc.1 (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))
Assertion
Ref Expression
necon2abiddc (𝜑 → (DECID 𝜓 → (𝜓𝐴𝐵)))

Proof of Theorem necon2abiddc
StepHypRef Expression
1 necon2abiddc.1 . . . 4 (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))
2 bicom 139 . . . 4 ((𝐴 = 𝐵 ↔ ¬ 𝜓) ↔ (¬ 𝜓𝐴 = 𝐵))
31, 2syl6ib 160 . . 3 (𝜑 → (DECID 𝜓 → (¬ 𝜓𝐴 = 𝐵)))
43necon1abiddc 2370 . 2 (𝜑 → (DECID 𝜓 → (𝐴𝐵𝜓)))
5 bicom 139 . 2 ((𝐴𝐵𝜓) ↔ (𝜓𝐴𝐵))
64, 5syl6ib 160 1 (𝜑 → (DECID 𝜓 → (𝜓𝐴𝐵)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  DECID wdc 819   = wceq 1331   ≠ wne 2308 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  df-ne 2309 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator