ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  necon2bd Structured version   GIF version

Theorem necon2bd 2257
Description: Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
Hypothesis
Ref Expression
necon2bd.1 (φ → (ψAB))
Assertion
Ref Expression
necon2bd (φ → (A = B → ¬ ψ))

Proof of Theorem necon2bd
StepHypRef Expression
1 necon2bd.1 . . 3 (φ → (ψAB))
2 df-ne 2203 . . 3 (AB ↔ ¬ A = B)
31, 2syl6ib 150 . 2 (φ → (ψ → ¬ A = B))
43con2d 554 1 (φ → (A = B → ¬ ψ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1242  wne 2201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-in1 544  ax-in2 545
This theorem depends on definitions:  df-bi 110  df-ne 2203
This theorem is referenced by:  nneo  8117  zeo2  8120
  Copyright terms: Public domain W3C validator