ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  necon2d GIF version

Theorem necon2d 2344
Description: Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
Hypothesis
Ref Expression
necon2d.1 (𝜑 → (𝐴 = 𝐵𝐶𝐷))
Assertion
Ref Expression
necon2d (𝜑 → (𝐶 = 𝐷𝐴𝐵))

Proof of Theorem necon2d
StepHypRef Expression
1 necon2d.1 . . 3 (𝜑 → (𝐴 = 𝐵𝐶𝐷))
2 df-ne 2286 . . 3 (𝐶𝐷 ↔ ¬ 𝐶 = 𝐷)
31, 2syl6ib 160 . 2 (𝜑 → (𝐴 = 𝐵 → ¬ 𝐶 = 𝐷))
43necon2ad 2342 1 (𝜑 → (𝐶 = 𝐷𝐴𝐵))
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:  map0g  6550  hashprg  10522
  Copyright terms: Public domain W3C validator