Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  necon2bbii GIF version

Theorem necon2bbii 2270
 Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon2bbii.1 (DECID 𝐴 = 𝐵 → (𝜑𝐴𝐵))
Assertion
Ref Expression
necon2bbii (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜑))

Proof of Theorem necon2bbii
StepHypRef Expression
1 necon2bbii.1 . . . 4 (DECID 𝐴 = 𝐵 → (𝜑𝐴𝐵))
21bicomd 129 . . 3 (DECID 𝐴 = 𝐵 → (𝐴𝐵𝜑))
32necon1bbiidc 2266 . 2 (DECID 𝐴 = 𝐵 → (¬ 𝜑𝐴 = 𝐵))
43bicomd 129 1 (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜑))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  DECID wdc 742   = wceq 1243   ≠ wne 2204 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  df-ne 2206 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator