MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  necon1d Structured version   Visualization version   GIF version

Theorem necon1d 2963
Description: Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
necon1d.1 (𝜑 → (𝐴𝐵𝐶 = 𝐷))
Assertion
Ref Expression
necon1d (𝜑 → (𝐶𝐷𝐴 = 𝐵))

Proof of Theorem necon1d
StepHypRef Expression
1 necon1d.1 . . 3 (𝜑 → (𝐴𝐵𝐶 = 𝐷))
2 nne 2945 . . 3 𝐶𝐷𝐶 = 𝐷)
31, 2imbitrrdi 251 . 2 (𝜑 → (𝐴𝐵 → ¬ 𝐶𝐷))
43necon4ad 2960 1 (𝜑 → (𝐶𝐷𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wne 2941
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 206  df-ne 2942
This theorem is referenced by:  disji  5132  mul02lem2  11391  mhpmulcl  21692  xblss2ps  23907  xblss2  23908  lgsne0  26838  h1datomi  30834  eigorthi  31090  disjif  31809  lineintmo  35129  poimirlem6  36494  poimirlem7  36495  2llnmat  38395  2lnat  38655  tendospcanN  39894  dihmeetlem13N  40190  dochkrshp  40257  remul02  41278  remul01  41280  sn-0tie0  41312
  Copyright terms: Public domain W3C validator