MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  necon2d Structured version   Visualization version   GIF version

Theorem necon2d 2961
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 2939 . . 3 (𝐶𝐷 ↔ ¬ 𝐶 = 𝐷)
31, 2imbitrdi 251 . 2 (𝜑 → (𝐴 = 𝐵 → ¬ 𝐶 = 𝐷))
43necon2ad 2953 1 (𝜑 → (𝐶 = 𝐷𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wne 2938
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 207  df-ne 2939
This theorem is referenced by:  map0g  8923  cantnf  9731  hashprg  14431  bcthlem5  25376  deg1ldgn  26147  cxpeq0  26735  lfgrn1cycl  29835  uspgrn2crct  29838  poimirlem17  37624  poimirlem20  37627  poimirlem22  37629  poimirlem27  37634  islshpat  38999  cdleme18b  40275  cdlemh  40800  prjspner1  42613
  Copyright terms: Public domain W3C validator