ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  necon1aidc GIF version

Theorem necon1aidc 2336
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
Hypothesis
Ref Expression
necon1aidc.1 (DECID 𝜑 → (¬ 𝜑𝐴 = 𝐵))
Assertion
Ref Expression
necon1aidc (DECID 𝜑 → (𝐴𝐵𝜑))

Proof of Theorem necon1aidc
StepHypRef Expression
1 df-ne 2286 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon1aidc.1 . . 3 (DECID 𝜑 → (¬ 𝜑𝐴 = 𝐵))
3 con1dc 826 . . 3 (DECID 𝜑 → ((¬ 𝜑𝐴 = 𝐵) → (¬ 𝐴 = 𝐵𝜑)))
42, 3mpd 13 . 2 (DECID 𝜑 → (¬ 𝐴 = 𝐵𝜑))
51, 4syl5bi 151 1 (DECID 𝜑 → (𝐴𝐵𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  DECID wdc 804   = wceq 1316  wne 2285
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 588  ax-in2 589  ax-io 683
This theorem depends on definitions:  df-bi 116  df-stab 801  df-dc 805  df-ne 2286
This theorem is referenced by:  necon1idc  2338
  Copyright terms: Public domain W3C validator