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

Theorem nemtbir 3028
Description: An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
Hypotheses
Ref Expression
nemtbir.1 𝐴𝐵
nemtbir.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
nemtbir ¬ 𝜑

Proof of Theorem nemtbir
StepHypRef Expression
1 nemtbir.1 . . 3 𝐴𝐵
21neii 2932 . 2 ¬ 𝐴 = 𝐵
3 nemtbir.2 . 2 (𝜑𝐴 = 𝐵)
42, 3mtbir 322 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1534  wne 2930
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 2931
This theorem is referenced by:  opthwiener  5520  opthprc  5746  ord2eln012  8527  snnen2oOLD  9261  0sdom1dom  9272  cfpwsdom  10627  fprodn0f  15993  m1exp1  16378  pmtrsn  19517  gzrngunitlem  21429  logbmpt  26816  sltval2  27686  sltsolem1  27705  nolt02o  27725  ex-id  30367  ex-mod  30382  coss0  38177  ensucne0  43196  clsk1indlem4  43711  clsk1indlem1  43712  etransc  45904
  Copyright terms: Public domain W3C validator