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

Theorem neorian 2794
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 2664 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
2 df-ne 2664 . . 3  |-  ( C  =/=  D  <->  -.  C  =  D )
31, 2orbi12i 521 . 2  |-  ( ( A  =/=  B  \/  C  =/=  D )  <->  ( -.  A  =  B  \/  -.  C  =  D
) )
4 ianor 488 . 2  |-  ( -.  ( A  =  B  /\  C  =  D )  <->  ( -.  A  =  B  \/  -.  C  =  D )
)
53, 4bitr4i 252 1  |-  ( ( A  =/=  B  \/  C  =/=  D )  <->  -.  ( A  =  B  /\  C  =  D )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    =/= wne 2662
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 185  df-or 370  df-an 371  df-ne 2664
This theorem is referenced by:  oeoa  7258  wemapso2OLD  7989  wemapso2lem  7990  recextlem2  10192  crne0  10541  crreczi  12271  gcdcllem3  14026  bezoutlem2  14052  dsmmacl  18639  txhaus  20014  itg1addlem2  21970  coeaddlem  22511  dcubic  23041  sibfof  28114
  Copyright terms: Public domain W3C validator