MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  neorian Structured version   Unicode version

Theorem neorian 2693
Description: A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
Assertion
Ref Expression
neorian  |-  ( ( A  =/=  B  \/  C  =/=  D )  <->  -.  ( A  =  B  /\  C  =  D )
)

Proof of Theorem neorian
StepHypRef Expression
1 df-ne 2603 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
2 df-ne 2603 . . 3  |-  ( C  =/=  D  <->  -.  C  =  D )
31, 2orbi12i 509 . 2  |-  ( ( A  =/=  B  \/  C  =/=  D )  <->  ( -.  A  =  B  \/  -.  C  =  D
) )
4 ianor 476 . 2  |-  ( -.  ( A  =  B  /\  C  =  D )  <->  ( -.  A  =  B  \/  -.  C  =  D )
)
53, 4bitr4i 245 1  |-  ( ( A  =/=  B  \/  C  =/=  D )  <->  -.  ( A  =  B  /\  C  =  D )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    =/= wne 2601
This theorem is referenced by:  oeoa  6843  wemapso2  7524  recextlem2  9658  crne0  9998  crreczi  11509  gcdcllem3  13018  bezoutlem2  13044  txhaus  17684  itg1addlem2  19592  coeaddlem  20172  dcubic  20691  dsmmacl  27198
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 179  df-or 361  df-an 362  df-ne 2603
  Copyright terms: Public domain W3C validator