MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  neeq2i Structured version   Visualization version   GIF version

Theorem neeq2i 2846
Description: Inference for inequality. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
Hypothesis
Ref Expression
neeq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
neeq2i (𝐶𝐴𝐶𝐵)

Proof of Theorem neeq2i
StepHypRef Expression
1 neeq1i.1 . . 3 𝐴 = 𝐵
21eqeq2i 2621 . 2 (𝐶 = 𝐴𝐶 = 𝐵)
32necon3bii 2833 1 (𝐶𝐴𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  wne 2779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-cleq 2602  df-ne 2781
This theorem is referenced by:  neeqtri  2853  suppvalbr  7163  disjdsct  28669  divnumden2  28757  nosgnn0  30861  upgr3v3e3cycl  41349  upgr4cycl4dv4e  41354
  Copyright terms: Public domain W3C validator