NFE Home New Foundations Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  NFE Home  >  Th. List  >  necon3d GIF version

Theorem necon3d 2554
Description: Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
Hypothesis
Ref Expression
necon3d.1 (φ → (A = BC = D))
Assertion
Ref Expression
necon3d (φ → (CDAB))

Proof of Theorem necon3d
StepHypRef Expression
1 necon3d.1 . . 3 (φ → (A = BC = D))
21necon3ad 2552 . 2 (φ → (CD → ¬ A = B))
3 df-ne 2518 . 2 (AB ↔ ¬ A = B)
42, 3syl6ibr 218 1 (φ → (CDAB))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1642  wne 2516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 177  df-ne 2518
This theorem is referenced by:  necon3i  2555  pm13.18  2588  ssn0  3583  pssdifn0  3611  uniintsn  3963  evenodddisj  4516  sfinltfin  4535  leltctr  6212  nnltp1c  6262
  Copyright terms: Public domain W3C validator