ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  necon2ai GIF version

Theorem necon2ai 2339
Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon2ai.1 (𝐴 = 𝐵 → ¬ 𝜑)
Assertion
Ref Expression
necon2ai (𝜑𝐴𝐵)

Proof of Theorem necon2ai
StepHypRef Expression
1 necon2ai.1 . . 3 (𝐴 = 𝐵 → ¬ 𝜑)
21con2i 601 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
3 df-ne 2286 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
42, 3sylibr 133 1 (𝜑𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = 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
This theorem depends on definitions:  df-bi 116  df-ne 2286
This theorem is referenced by:  necon2i  2341  neneqad  2364  intexr  4045  iin0r  4063  tfrlemisucaccv  6190  pm54.43  7014  renepnf  7781  renemnf  7782  lt0ne0d  8243  nnne0  8716  nn0nepnf  9016  hashennn  10494  bj-intexr  13033
  Copyright terms: Public domain W3C validator